It depends on where the conversion took place in the Ottoman Empire and the individual converting. Of course, there are the Islamic claims that hold that Christians discovered that Islam was the correct religion and desirable for society, leading to conversion However, there are reasons for conversion from Christianity to Islam away from the merits.
Forcible Conversion
In the Ottoman Empire there was the practice of Devshirme, ripping innocent Christian boys and girls away from their families to serve the Sultan and be forcibly converted Islam. The parents would have no choice as to whether their children would be deprived from them at the tender age of five years. It was a tax. The women would grow up to be part of the Sultan's harem. While they served the Sultan, each woman could hope that her son would be chosen as the Crown Prince and she could have court rights, but she could never hope to have rights to her own body. The men were educated in the arts of war and diplomacy. The ones of who were more physically capable would become the feared shock-troops of the Ottoman Empire: the Janissaries. Those more skilled in diplomacy would be the ministers of the Sultan's court. Aside from a serious incident of forcible conversion, the practice of Devshirme is a clear violation of a person's inalienable right to family.
To Escape Dhimmi Status
Non-Muslims who lived in the Ottoman Empire were called Dhimmi, or second-class citizens. They lacked a number of fundamental rights such as freedom of movement, freedom of settlement, access to all occupations, and additional taxation because they were Non-Muslim. Each Dhimmi group, like the Armenians, Greeks, Jews, and so forth were organized into Millets which were community representative groups. Each Millet would report to a centralized Millet representative in Istanbul who would have direct dealing with the Imperial Palace. This would allow the Millets to air their grievances, but was more often used as a method to control the Millets and make them less resistive to Ottoman Occupation.
Admittedly, life under the Ottomans was far superior to most contemporaneous Empires. Spain was by far a more religiously and culturally intolerant place during this period as were France and England.
However, if such a person converted to Islam, all of these inequalities would drop away. Islamic States have historically created a set of benefits that only Muslims could partake, making a choice to be Muslim tied to choices about economic viability (which trades you could work - especially in government) or tied to choices of social standing. These would strongly weight the choice and produce a steady stream of converts by their very nature.
Leadership
Most positions in the Ottoman political leadership were restricted to Muslims only. This encouraged conversion as a method of gaining more prestigious government offices.
No religion "inspired" the Ottoman Empire. The Ottoman Empire had Sunni Islam as its State Religion, but it was not a view or tenet of Islam to create an Ottoman Empire.
Nothin Islam is still around
The Caliph of the Ottoman Empire represented the religious leadership of the Sunni Muslims and granted legitimacy to the Ottoman Empire as the vanguard of Islam.
Turkey was the center of the Ottoman Empire, and its values were based in Islam.
Certain political and economic benefits were reserved for Muslim in the ottoman empire
No religion "inspired" the Ottoman Empire. The Ottoman Empire had Sunni Islam as its State Religion, but it was not a view or tenet of Islam to create an Ottoman Empire.
Sure, there are Christians who converted to Islam. Refer to the link below.
NO. The Safavids were a rival empire to the Ottomans. The Janissaries were the elite troops of the Ottoman Empire, usually made of Serb, Greek, and Bulgarian Christians who were forcibly converted to Islam and made to fight against all of the Ottomans' enemies in exchange for perceived power.
Islam
Nothin Islam is still around
The Caliph of the Ottoman Empire represented the religious leadership of the Sunni Muslims and granted legitimacy to the Ottoman Empire as the vanguard of Islam.
The Ottoman Empire.
Turkey was the center of the Ottoman Empire, and its values were based in Islam.
Certain political and economic benefits were reserved for Muslim in the ottoman empire
The Ottoman Empire held that SUNNI ISLAM as understood by the Caliph in Istanbul was the official religion. However, there were many religious minorities throughout the Ottoman Empire. Most important among these were the Orthodox Christians (Greeks, Jugoslavs, Georgians, and Ukrainians), Armenian Christians, Catholics (Hungarians and Jugoslavs), Coptic Christians (Egyptians and Nubians), and Jews (mixed throughout). There were also smaller religious minorities like the Sabeans, Shabaki, Mandaeans, Yezidis, Yarsanis, Baha'i, and others in specific areas of the empire.
The underlying premise of the question is incorrect. The Byzantine Empire never converted to Islam and Islam was, by and large, illegal within the Byzantine Empire for as long as it existed. The reason why the territory formerly controlled by the Byzantine Empire became Muslim-majority was because it was conquered by various Islamic States such as the Rashidun Caliphate, the Umayyad Caliphate, the Seljuq Turkish Empire, and the Ottoman Empire (which finished the job).
Islam. Peace.