Do you know the history of the Bosnian bogomils?
Since the late 12th century, Bosnia was a stronghold of bogomils. These were dualist Christians who held a faith similar to the French cathars, they believed all material things were created by the devil, who also created the world, and trapped human souls which belong to God into bodies of his creation. Their belief was similar to the earlier gnostic christians, the Armenian paulicians (who settled in Thrace and spread the religion from there all over Europe) and the manicheans. Even the Church of Bosnia, originally catholic, became a bogomil (also known as patarene) institution. They believed themselves to be the original christians and called themselved krstjani. They refuted the cross, baptism, marriage, the eucharist and church hierarchy. They only knew credentes, believers, who were free to live as they wanted as long as they did not harm others, and perfecti, who lead a very ascetic life as to invite the Holy Spirit in them. Most believers only became perfects on their death-bed.
Until the Ottomans came, the heretic Bosnians were ruthlessly persecuted by the Roman Catholic Church. In fact, the RCC found the Inquisition to root out this type of heresy, in which they succeeded in France with the cathars. Catholic Hungary had been attacking Bosnia for the last couple of centuries, and as a result, being catholic did not make you very popular in Bosnia. The bogomils had invited the Turks to protect them from this persecution.
The Bosnian muslims are for the most part direct descendants of these bogomil heretics, who converted to islam over a period of 200 years. The Ottomans brought previously unseen prosperity to Bosnia, cities were founded and the economy was flourishing. They initially forced the Bosnian noblemen, all bogomils, to convert to islam if they wanted to keep their possessions, and their sons were taken to Turkey and returned as jannissaries, but the other classes were left at peace.
Also, from the Ottoman administration we see that the heretic christians were treated favorably by the Ottomans, who called them kristians (while Orthodox and Catholic christians were called kuffar). In the early 17th century the term kristian disappears, and Bosnia is predominantly muslim (around that time another heresy appears, islamic in name this time, but similar to bogomilism, but is destroyed by the Ottomans very fast).
We can assume that not every christian heretic was very happy with the advent of islam (even if sufism was the most popular, again a heterodox branch of religion), and it is known that a minor part of the bogomils converted to Orthodox christianity.
