Looking at most AH scenario's and timelines, it strikes me that if there is a Turkish Muslim nation/empire in Anatolia, then it is either the Seljuk Rum Sultanate or the Ottoman Empire, or occasionally something that's just called the Seljuk Empire...
Now then, both the Great Seljuk Empire as well as its most successful successor state, the Rum Sultanate, suffered a lot from internal conflicts and dynastic struggles, and it was pretty much inevitable that both nations would fall apart at some point, and therefore it is only natural that, at some point, one of the many Turkish successor state would rise and become at least a local power.
And the Osmanoglu beylik that eventually grew out into the Ottoman Empire was actually one of the smaller and weaker beyliks, and around the year 1300, there was quite a number of stronger and more powerful beyliks at that time such as Germiyan and Karamanoglu...
..
Now then, here's the challenge, should you accept it; create a Turkish Empire that has developed out of a successor state of the Rum Sultanate or at least the Great Seljuk Empire, other than the Osmanoglu beylik.
The minimal requirement is that this state is a stable, local power that is not at serious risk of being overtaken by a neighbouring state or empire.
Bonus points of your scenario has two or more such Turkish states.
...and, of course, bonus points for taking Constantinople.
Now then, both the Great Seljuk Empire as well as its most successful successor state, the Rum Sultanate, suffered a lot from internal conflicts and dynastic struggles, and it was pretty much inevitable that both nations would fall apart at some point, and therefore it is only natural that, at some point, one of the many Turkish successor state would rise and become at least a local power.
And the Osmanoglu beylik that eventually grew out into the Ottoman Empire was actually one of the smaller and weaker beyliks, and around the year 1300, there was quite a number of stronger and more powerful beyliks at that time such as Germiyan and Karamanoglu...
..
Now then, here's the challenge, should you accept it; create a Turkish Empire that has developed out of a successor state of the Rum Sultanate or at least the Great Seljuk Empire, other than the Osmanoglu beylik.
The minimal requirement is that this state is a stable, local power that is not at serious risk of being overtaken by a neighbouring state or empire.
Bonus points of your scenario has two or more such Turkish states.
...and, of course, bonus points for taking Constantinople.