Turkey is the only country in the world with a majority Muslim population but a secular government (which must remain secular by Constitution). Last I read about it, a large, large majority of the country wants to keep it secular. But, as you semi-mention, there are muslim political groups in Turkey, including the ruling AKP party. It's been getting a bit authoritarian recently and may be a threat to secular freedom.
In many ways they're more secular than we are. Not only are head scarves banned in public buildings but the current Prime Minister Erdogan, once spent 6 months in jail for reciting a religious poem in public, and was also removed from office as mayor of Istanbul. So despite the fact that Turkey is 95% Muslim, Muslims have more freedom to practice their religion in the way that they choose in the U.S. than they do in Turkey.