Carson spent three seasons with West Brom after joining from Liverpool, where he had been a frustrated reserve.
Albion manager Roy Hodgson said: "In the three months I worked with Scott I cannot fault his professionalism and dedication.
"He has been a good servant to the club and I would like to wish him all the best for the future as he looks to broaden his experience by playing in a foreign country."
In total he made 118 appearances for the club and sporting and technical director Dan Ashworth said: "I would like to thank Scott for his efforts and contribution to the last three seasons.
"After the disappointment of being relegated from the Barclays Premier League he played a significant part in our last two successful seasons.
"It is a good opportunity for Scott to go to Turkey and on behalf of everyone at the football club I would like to wish him well in his future career."