While it is true that the relationship between the Catholic and Orthodox churches did decline this isn't them main reason for the 4th Crusade conquering Constantinople. The 4th Crusade wasn't called against the Byzantine Empire, infact the pope excommunicated the participants of the 4th Crusade because they weren't fighting against the target the pope had intended.

The 4th Crusades' original target was Egypt, the Crusaders planned to capture Egypt, and then move on to the holy land, in order to prevent a counterattack from Egypt against the Holy Land when the Crusaders left, as happened after the 1st and 2nd Crusades.

The Crusaders decided to travel by boat because they were heading to Egypt, and because previous expeditions by the German and French contingents that travelled by land through Anatolia during the 2nd and 3rd Crusades had suffered high casualties.

Therefore the Crusaders commissioned Venice to build a fleet for them, promising to pay them 85,000 marks when the crusade began, however when the Crusaders showed up they only had 35,000 marks, and since Venice had made a huge investment in building the fleet they weren't willing to reduce their price by that much.

Instead Enrico Dandolo, who was the Doge of Venice said that they would transport the crusaders, if the crusaders would help the Venetians recapture the city of Zadar(also known as Zara), which had rebelled against Venetian rule in 1181. Doge Enrico also prevented food from being transported to the island the Crusaders were stationed on until the crusaders agreed to the deal. Therefore the crusades' leaders agreed, despite the fact that Zadar was a Catholic city, and under the protection of Emeric I, King of Hungary and Croatia.

In October 1202 the Crusade left Venice and they besieged and captured Zara in November 1202, in response to this King Emeric asked Pope Innocent III to excommunicate the Crusaders, which he did, although most members of the Crusade didn't hear about this fact.Since the siege had ended on November 24, by which time the weather wasn't suitable for sailing, the Crusade had to spend winter in Zara.

Meanwhile Boniface I Marquess of Montferrat who was one of the leaders of the Crusade had left to visit his cousin Philip the Duke of Swabia, and King of Germany. Philip was married to Irene. Irene had a brother, Alexios IV (who wasn't emperor at this point I'm just including his regnal number to avoid confusion) who was living at their court. The father of Alexios IV was overthrown imprisoned and blinded in a coup carried out by his older brother Alexios III, in 1195. Ever since this Philip had supported Alexios IV claim on the Byzantine throne and they plotted to place Alexios IV on the Byzantine throne. When Boniface went to Swabia he met with Alexios IV, and they presumably reached an agreement, although we don't know exactly what happened at their meeting due to a lack of primary sources.

In January 1203 Boniface arrived at Zara alongside envoys from Alexios IV, Alexios IV offered to give the crusaders 200,000 marks, as well as pay off all of their debts to the Venetians, give 10,000 men to the crusade and permanently maintain 500 knights in the holy land, have the Byzantine navy transport the crusaders, and mend the great schism of 1054 by placing the Byzantine Church(Eastern Orthodox) under the authority of the Pope. This offer was strongly supported by Boniface, Doge Enrico, and Louis I Count of Blois. The other leaders of the crusade eventually agreed to the offer as well. This prompted a few of the crusaders to leave the Crusade, however the vast majority of the crusaders remained with the crusade.

When the weather improved and the crusade gathered supplies, in April 1203 the crusade left Zara alongside the Venetians, at this point a majority of the Crusading force was Venetian. With there being 14,000 Venetians and 10,000 "regular" non-venetian crusaders. The Venetians relationship with the Byzantines was much worse than that of other Catholic countries. Since Venice made a large amount of its wealth by trade, and the Byzantines were rivals to the Venetians, and through their competition and their favourable treatment to Byzantine merchants were harming Venetian profits. Also in 1182 after Alexios II was overthrown by Andronikos I, the supporters of Andronikos I performed a massacre of Catholics who lived in Constantinople, which resulted in over 10,000 Catholics dying (there is a wide number of estimates for the death toll ranging from 10,000-80,000). Before 1182 there were a large number of Venetians living in Constantinople, and many soldiers in the Venetian navy had lost relatives in the massacre, causing the Venetians to hate the Byzantines a lot more than average Catholics hated them.

In June 1203 the crusade arrived in Constantinople . The Crusaders had hoped having Alexios IV with them would convince the garrison to defect, but that didn't happen. During the siege in July Alexios III, after a failed sally attempt fled the city prompting the people of Constantinople to release Isaac II from his imprisonment and proclaim him as Emperor, the Crusaders then demanded that Alexios IV should be proclaimed as Emperor as well, at which point they stopped besieging the city.

At this point the Crusaders didn't control Constantinople, instead it was in the hands of their ally Alexios IV. In August 1203 another smaller massacre of Catholics happened in Constantinople, due to the Crusader army being away fighting against Alexios III in Thrace. Meanwhile Alexios IV was trying to fulfil the promises he had made to the Crusaders. Placing the Eastern Orthodox Church under the Pope was very slow, however even this little effort annoyed a great many Byzantines. The main promise Alexios IV tried to fulfil was the financial one, however even after melting down statues to make money he could only raise 100,000 of the 300,000 marks he needed to make the payment.

His attempts to fulfil his promise to the Crusaders made Alexios IV very unpopular with the people of Constantinople, who overthrew and killed him and his father on February 1204, and named Alexios V as Emperor instead. The crusaders still demanded that the promise of Alexios IV should be fulfilled, however when they realised that Alexios V wasn't going to abide by the deal the Crusaders besieged Constantinople, sacking it in April 1204.

In summary political upheaval in the Byzantine Empire resulting in the massacre of Catholics soured the relationship between the Byzantines and Catholics, leading to the Venetians hijacking an indebted Crusading force and taking it to capture Constantinople. The pope had called the 4th Crusade against Egypt, not the Byzantines.

u/wowbuggertheinfinite has a great discussion of the Fourth Crusade and how it all went so off the rails, but I want to highlight another important part of this whole incident: relations between the Latin West (and Crusaders in particular) and Byzantium were on rocky footing well before the Fourth Crusade.

Relations between Byzantium and the West deteriorated significantly in the Central Middle ages. One major contributing factor was the Norman 'conquest' of southern Italy. I say 'conquest', because it was more like a slow edging out of the previous political elites and replacing them with newly arrived Normans (initially hired as mercenaries) than an out and out brutal conquest, and in that way very different from the more famous Norman Conquest in 1066 (which is not to say it was entirely peaceful either, it was still pretty violent). Byzantium ostensibly ruled over southern Italy and the previous rulers had sworn allegiance to the Empire. The new Norman nobility were less interested in paying their respects to a distant power and Byzantium lacked the strength to force them to, especially after the disastrous defeat at Manzikert in 1071.

Things took a significant turn for the worse in the 1080s when the Normans, led by Robert Guiscard a prominent leader in the conquest of Southern Italy and Sicily, invaded the Balkans - lands definitely under Byzantine rule. This conflict lasted from 1080 until Guiscards death in 1085 - which roughly coincided with a major Byzantine-Venetian victory against the Norman forces. Somewhat ironically, many of the Norman participants in this invasion would go on to join the First Crusade, most notably the future Prince of Antioch (and Robert Guiscard's son) Bohemond of Taranto.

The First Crusade had a rocky start to Byzantine-Crusader relations (the presence of his Norman former enemies was not exactly something Alexios I loved), but mostly everyone got along reasonably well until the siege of Antioch. This is one of the major events of the First Crusade, and literal books have been written about it, so I'm going to limit myself to the big picture and skip over many very important and complex details. The short version is that the Crusader forces pushed ahead of Alexios' armies and besieged the stronghold of Antioch, at the same time the Atabeg of Mosul, Kerbogha, was summoning a massive Muslim army to crush the Crusade. The siege of Antioch dragged on for months, things were looking dire for the Crusaders. Bohemond managed to get a local resident to betray the city and let in a small force one night, and the Crusaders took the city just in time for Kerbogha to surround it and put them under siege. Alexios was supposed to come relieve them, he didn't (much ink has been spilled over this decision), it looked like the Crusaders were all about to die, but in a rather miraculous turn of events they road out of the city to fight Kerbogha and won. The First Crusade was triumphant, and feeling like they were betrayed by Alexios and the Byzantines. Bohemond was given control of Antioch, and declared that he didn't have to turn it over to the Byzantines despite previously have agreed to do so because they had abandoned the Crusade in its time of need. Alexios didn't really see things that way.

The relationship between Byzantium and Antioch would continue to be fraught for the next two centuries. Bohemond actually assembled another army while touring Europe promoting his success on the crusade and used it to invade Byzantium again, this time with disastrous consequences. He was roundly defeated, captured by Alexios, and forced to pledge his loyalty to the Byzantine Empire. He died without returning to Antioch, and his son (now Prince of Antioch), decided that his father's oath to Alexios didn't apply to him. Relations continued to be problematic after that.

Relations between Byzantium and the Latin West continued to be problematic through the Second Crusade, although they again mostly cooperated but there were a lot of hard feelings after the Crusades failure and a strong desire in parts of the West to blame that failure upon the 'treachery of the Greeks'. Relations took an even worse turn in 1185 when the Angelos dynasty deposed the last Komnenoi emperor and then pretty much refused to help with the Third Crusade at all. The Angelos dynasty was marked in particular by strong anti-Latin sentiments, and were eventually deposed as part of the Fourth Crusade.

This has been a super broad survey of a very complicated topic, but I think it is important to keep in mind that throughout this period relations between Byzantium and the Crusaders were not all rosy, and while the Fourth Crusade is arguably an extreme result, it is not entirely without precedent.

Peter Frankopan's A Call from the East is a good history of the First Crusade from the Byzantine perspective and covers the fraught relations beforehand pretty well.

Thomas Asbridge's The Crusades is a great survey of the whole period.

Andrew Buck's The Principality of Antioch and its Frontiers in the Twelfth Century is a serious scholarly work, and not one I'd recommend a beginner jump in to, but has some great detail on the complicated relationship between Antioch and Byzantium in the century before the Fourth Crusade.