I think that it is important to understand the initial reasons for the First Crusade (that being Alexios I Komnenos' request for martial aid to Pope Urban II) as well as the wider goals of the Crusaders throughout the first seven Crusades, each having the ultimate goal of capturing Jerusalem. The crusades that targeted Egypt did so because that was the seat of the Ayyubid Islamic Dynasty. Egypt would also serve as an important launching point to Jerusalem, it being a massive bread basket in the region, thus allowing the Crusaders to field large numbers of zealous men. No matter what was executed during these crusades, their goal was the holy city. North Africa was at the time a poor region far away from the Levant that had little that could benefit men on a Crusade. The era of Carthage had long since passed, and North Africa held little strategic value and, much more importantly, next to no religious value to the Christians.

Crusading warfare has a number of advantages, as well as limitations.

It let's you assemble an army that draws on the manpower of multiple states, while also joining them in a zeal that makes them face terribly adverse odds and willingly undergo hardships that would make even the toughest mercenaries pack and go home.

At the same time, this army has one objective, and one objective only: Jerusalem, the Holy City. Jerusalem, the city where Jesus died and rose from the dead. Jerusalem, the capital city of the Holy Land, where Jesus was born and where he worked his miracles. This city is so central to Medieval Europeans' understanding of the world and their place in it that they put it in the centers of their maps and their own countries on the fringe.

In the mind of the common soldiers, there was no question, it had to be Jerusalem. Anything else was a distraction. It might be a distraction conveniently on the way, full of loot and strategically compelling, if your ruler said so, but ultimately it was just a side dish. The 1st Crusade did conquer Antioch, but when its leaders (who had more opportunistic ideas) tried to seize other cities as well, their soldiers almost mutinied, because they wanted to continue with their quest. Edit: Many crusaders understood their campaigns as "armed pilgrimages" rather than wars of conquest.

This single-minded determination would ultimately cost the crusaders dearly. E.g. during the 3rd Crusade, Richard the Lionheart advocated to capture Egypt and thus put an end to Saladin. But so close to their goal, his soldiers would have none of it, which lead to a stalemate. Also, even capturing Jerusalem was not necessarily a „Happy End“. The city was strategically very exposed, not easily defensible, and the Kingdom of Jerusalem was dwarfed by its Muslim neighbors (e.g. Egypt) in terms of population. Once the city was captured, most crusaders would have just returned home, leaving it to fend for itself.

Later crusades tried to capture Egypt first, just as you said, but only in order to pick up Jerusalem later, once the big. rich, dangerous Muslim juggernaut was taken care of. But this in part reflected their different nature. E.g. the 7th Crusade was effectively a military expedition of the Kingdom of France, lead by its very devout king, not an ad-hoc affair like the 1st Crusade. For this reason, it could place strategic interest over religious zeal and attack Egypt first – in order to take Jerusalem more easily later.

HOWEVER:

There were Christians who DID attack Tunisia (aka „Carthage)“. The Normans who captured Siciliy in the 11th century had further imperial ambitions and Tunisia was conveniently close, so they tried to expand there. However, this was not a „crusade“ but just a „selfish“ endeavor of a Norman dynasty. And ultimately, the African possessions of the Normans were reconquered by African states. On a side note: Some people did not trust the Sicilian Normans during the First Crusade for similar reasons, because they thougth that they joined for empire bulding, rather than true devotion.

Also, there was a late Crusade directed against Tunisia, but it did not amount to much and after its demise, the crusaders that did not just return home to Europe tried to reach Jerusalem instead.

So in total, the Crusaders did not capture North Africa, because they did not greatly care about fighting Muslims, or rolling back Islam, or reconquering Roman territory, or about conquering new territories in distant lands instead of seizing their neighbors' fiefs. They cared about Jerusalem. And Jerusalem is not in Northern Africa. And on the few occasions when they did venture into Tunisia, they were defeated.

Sources:

Asbridge: The Crusades

Minor spelling edits

Your subject headline question is workable and worth looking into. Yet, I would consider narrowing your specifics. You mention Rome, and the fourth crusade, as well as general "time of crusades" in your paragraph.

This is a great question that I think can be answered through narrowing your search. Broad questions bring too many possibilities, too many opinions.

Edit: I'm drinking and I'm a nervous writer.