As Timelord64 referenced in his comment, it appears as though the Templars had infiltrated the Byzantine Empire after being defeated by the Assassins in Italy.
During the early 16th century, after the Templars were driven from Italy by the Assassins, the members of the Templar Order were able to seize control of the remnants of the Byzantine Empire.
Furthermore, the main Templar aim after taking control of the Byzantine empire was to conquer the neighbouring Ottomans.
Now in command, the Templars set up their primary headquarters at
Derinkuyu in Cappadocia, where the Byzantines' ultimate aim was to
uproot the Ottoman Empire and retake control of Constantinople.
This alone could prove reason enough for the Assassins to join the Ottoman Turks. If the Templars had taken over the Ottoman empire, they would have regained control of Constantinople, and in-so-doing gained a serious stronghold from which they could spread their ideology.
However, there was also an already established relationship between the Turks and the Assassins. Originally created by the Polo brothers Niccolo and Mafeo, an Assassin branch had been established in Constantinople long before the events in Revelations (~1200CE vs ~1500CE).
With this branch established, the assassins were able to help the Turks in a number of ways, including:
- fighting Spanish Templars posing as refugees during the Spanish Inquisition, and
- helping to end the Venetian-Ottoman war
Thus we can also see that the Assassins have been historically allied with the Ottomans, providing another reason why the Assassins might support the Turks over the Byzantines.
Regarding Selim I more specifically, it is important to remember that Selim was fighting to retake Constantinople from his brother Ahmet, who was the leader of the Byzantine Templars. Although Selim may have wanted to conquer and spread the power of his empire, having an ally of the Assassins in such a role would be far preferable to the Assassin order than having a Templar leader in such a role (though this inference is primarily based on my opinion). Further, Selim's son had high respect for the Assassins, a possible indication that the Ottomans would stay loyal to the Assassins in future generations.