How did the Osmanli Turks come to conquer the Byzantine Empire?
The Osmanli Turks (the Ottoman Turks as Anglicised by Europe) were a driving force in the Middle East over almost 700 years, starting in 1299. They first took power from the Seljuk Turks, who had ruled many countries from the Medieval Period, and were in decline following the First and Second Crusades.
King Osman I first moved the borders of Ottoman towards the Byzantine Empire, which itself was in decline, and he was able to take control of many lands through military victories; the Byzantines had been sacked by the Crusaders, and the once-mighty Seljuk Empire was occupied with fighting the Mongols in the East. Osman pledged his allegiance to the Seljuks to gain ground, but eventually grew his power to take control of their land as well. Osman's victories allowed displaced, transient, and wealthy Turks to move into the Byzantine lands, where they established a cultural foothold, and soon before his death Osman's forces won a crucial battle at Bursa, allowing them to wage war against Constantinople.
Constantinople was eventually captured by Mehmed II, who repurposed it as the center of the Ottoman Empire in 1453. This marked the formal end of the Byzantine Empire and the reign of the Ottoman Empire. Wars in the region continued between various factions, including the rise of the Kangxi Emperor in China and the unification of the Chinese people, but the Ottoman Empire persisted, both gaining and losing ground before finally dissolving in the Turkish War of Independence (1919-1922).