I cannot directly answer your question, but I can give you some related backstory:

Most of the world's Shiites lived in Persia/Iran after the forceful conversion of Safivid Iran to universal 12er Shiism (as opposed to a mix of sunni/shia) by the Safavid emperors during the 1500s onward. The Ottoman empire was the major Sunni stronghold and the vast majority of its populace was indeed Sunni; hence, the Safavids and Ottomans became major rivals throughout the centuries.

Today the effects of this are still seen today, with the major hotspots of Sunni/Shiite sectarianism being either former battlegrounds of hotly contested "frontier" territories of Iranian/Ottoman wars such as in Mesopotamia/Iraq; or, like in the situation of Bahrain, in territories that belonged to Iran before European colonial interference.

Also, you mention "Kurds" as if they are some other muslim sect -- they are an ethnic group, not a sect. Kurds are one of many different Iranian peoples who are majority Sunni who have traditionally lived in Kurdistan -- a region now divided between Iran, Turkey, Syria, and Iraq.

The Ottomans handled diversity remarkably well. Although the rulers were Sunni Muslims, Christians and Jews ran their own affairs in what was known as the millet system. This system broke down in the 19th century because of the development of nationalism.

As for different ethnic groups of Muslims, the Ottomans tried to minimize difference. In the early 20th century, the Young Turks had a mild pro-Turk ideology, and this existed alongside developing Kurdish, Arab, and Albanian national identities. But the policy for the preceding centuries was, basically, any Muslim regardless of ethnic background could function in the government. Usually this meant learning Turkish too, but not always. In the 19th century they were more particular about using Turkish as the governing language, so that everyone was on the same page. However, some administration was done in Arabic or Kurdish anyways. What Western observers would call a "Turk" in this period would be just as likely to be someone who was raised speaking Albanian, Circassian, Bosnian, Arabic, or Kurdish as Turkish.

Shiites are an interesting case. Under the Ottomans, Shiites in Anatolia, Lebanon, and Iraq were vastly in the minority compared to Sunnis. The state officially was against Shiism, but in the later period (19th, 20th centuries) its not clear how much this actually mattered. It seemed as long as you stayed quiet, and just acted Sunni in public, you could join the military or administration, and no one would care too much.

Problems came after the end of the empire. Its incorrect to just say the British and French broke it up without paying attention to ethnicity/religion, because honestly, the Ottomans didn't pay too much attention to these things when creating administrative units. There were two big changes: the first is obviously that the Ottoman state as a whole had a huge Sunni majority. All of a sudden, there were two new states, Lebanon and Iraq, with either Shiite majorities or much larger minorities. While the Ottomans, in the late phases of their empire, mostly ignored Shiites, now a historically marginalized group had to be taken seriously as a political force. The Anglo-French tendency to use "divide and conquer" tactics in colonial rule exacerbated this.

The other big change was the death of an inclusive "Ottmanist" ideology. The Ottoman state ideology was to try and unify the disparate groups as much as possible. Now, smaller states abandoned that and created individual nationalist ideologies. In Turkey, this meant changing from Muslim unity to Turkism. Now everyone in the Republic was forced to identify as Turkish. For smaller communities this meant assimilation. For the huge Kurdish population, it meant rebellion and creating their own strong nationalist response.

TL;DR The Ottomans tried to lessen differences between Muslims as much as possible, and were mostly successful. Turkish, British, and French policy after WWI exacerbated differences instead.

Before the collapse of the Ottoman empire there was less conflict in general (at least that I know of). Many of the more recent problems stem from the fact that after the First world war the Allies divided up the land based around (basically) arbitrary lines that made some sense on the map.

These decisions did not take into account the locals and their religious, ethnic, or historical differences. This has lead to places like Iraq where there are a sizeable majority of one group (Sunni Muslims) and a minority (Shiite).

Another good example of this is the Kurdish diaspora where they live in a relatively concentrated area but are not a nation state but separated within several nation states.