Unlike other countries that conduct
painful and torturous exams test for "chronic homosexuality" 
Turkey is not on the list.
I've had one visit to Istanbul (also traveling solo) and absolutely loved it. Didn't want to leave. People were friendly, easy to use public transit, and absolutely an abundance of gay life. Not to mention the sights and sounds throughout.
However, given your short visit, and assuming you're up for it:
I'd hustle over to a true
no-frills Turkish Bathhouse. They're in abundance, so search online first for some top pics. I've now forgotten the one I went to, but definitely a fascinating cultural experience. Plus you will feel re-energized, with what feels like an entire layer of skin removed from your body during the deep cleanse.
Good stuff.
Connecting with gay locals seems challenging in such a short amount of time.
Then again, if you have access to the TK lounge, you may not want to leave the airport at all =) Entire thread dedicated to this lounge