Turks conquered the byzantine empire because they had to?
Sometimes honesty is the best remedy. The Turks then where no less power hungry than the romans for example, and the byzantine people payed the tab. Somehow reshaping such horrible events into something good and justified doesn't seem right.
The key is the real peace with Islam
You know, in those times, for example in England, if you were no catholic, they would burn you. But in Islam, that is not allowed. Everyone lives what they want, so Ottomans are mainly muslims but they keep orthodoxs, catholics, jews, indian religioners in their people. Especially Armenians were the best people that is no-muslim to the Ottomans. But please dont take the issu to Armenian-Turkish things. Big Powers show Turkish people as enemy of Greeks, Armenians, Egyptians, Syrians, Georgians, Kurds, Bosnians, Serbians, Albanians, Hungarians but no, we are not hostile to those people. We lived with them in peace for 600 years. Big Powers do that cos there is still a possibility to build Ottomans in future. Those nations lived their best national welth when they were in Ottomans. If you can research on Ottoman bureaucracy records, you can see how much Ottomans spent for those nations. These records still exist. So, peace be with you aka Selamun Aleykum