1) The oldest and probably best option is Turkish Airlines. You can go wherever you want with it. Argentina, Belgium, Brazil, China, Congo, Denmark, France, Finland, Germany, Greece, India, Israel, Japan, Norway, Pakistan, Qatar, Russia, Ukraine, United Kingdom, United States, Zimbabwe are only a part of its destinations in the World.
2) AnadoluJet is fully owned subsidiary of Turkish Airlines and its headquarter is in the capital, Ankara. It has only two international destinations: Arbil in Iraq and Ercan (Nicosia) in TRNC.
3) SunExpress is a joint venture between Lufthansa and Turkish Airlines. Especially if you will travel from Germany, the company can be a good solution for you. You can prefer it for Denmark, Switerland, Lebanon and some more countries.
4) If you plan to visit İstanbul, Atlasglobal can be good idea. Some of its destinations are Albania, Armenia, Bosnia Herzegovina, Germany, Israel, Kuwait, Morocco, Norway, Russia, Saudi Arabia, Ukrania and United Kingdom.
5) Another company is Onur Air and it has flights from Germany, Iraq, Kazakhstan, Kuwait, Macedonia, Russia, Switzerland and Ukraine.
6) Pegasus is another one and its destinations are mostly European cities such as Amsterdam, Athens, Barcelona, Venice, and Vienna.