Why would we have to do anything first of all and second why would we need to gain legitimacy in the Muslim world?
Turkey is a country, a secular country. We do (or did) politics because they serve the interest of the Turkish people, whether they are Muslim, Atheist or Christian - Turk, Kurd or Armenian origin. That's the one thing you should understand. NATO is good for Turkey and we're good to NATO. The west or lets just say the EU is Turkeys biggest trade partner and investor by far. We export most of our goods there and import most of our goods from there. Why on earth would we ever dissassociate us from that?