Majority of the movies from English speaking countries are screened with Turkish subtitles, with the exception of movies appealing to children. Even for them it is possible to find screenings in the original language.
Most of the large scale film festivals such as Istanbul International Film Festival (in April), Film Ekimi (in October) or IF Independent Film Festival (February) show movies in other languages with both English and Turkish subtitles.
If you are interested in viewing recent Turkish movies in a theater, currently Katadromart , a non profit organization, holding Turkish movie screenings with English subtitles. For more information you can check their website.http://www.katadromart.com/en/
If you are fine with watching them home, iTunes Turkey has got a good collection (they are working with Turkish Film Channel which distributes Turkish films worldwide).