Those concepts like “West” or “Muslim World” are absolutely idiotic and oversimplified considering how many different nations are put in the same basket.
What does a Greek have in common with a Dutch or a Canadian? Or what about a Tajik with someone from Somalia or a Bosniak with someone from Bangladesh or Indonesia... other than a common religion (needless to say how many different schools and branches and understandings on it you have etc) they have absolutely nothing else in common.
The so called Muslim world is too big and diverse for it to have a single state that leads it. It neither has a state with the financial, military and soft power to do so. Most Muslim majority states are a mess and should first stop complaining about the evil Westerners sabotaging them and concentrate on improving and development.
Turks from the Western parts of the country have much more in common with the people of the Balkans and the Mediterranean than they do with Arabs or Afghans or Malays etc. On the other hand many people living closer to Syria or Iraq have more in common with the people living on the other side of the border than they do with the people of the Balkans or the Mediterranean. People living around Kars, Ardahan, Artvin etc have more in common with the Caucasus nations etc and etc.
Turkey is and have always been a mix and a bridge between the East and the West and is a very diverse country. Turkey is a part of Europe and the Balkans and it is a part of the Middle East, a part of the Caucasus region, a part of the Black Sea and Mediterranean, a part of the Turkic World no matter if you guys like it or not.