Hi Diana,
we are planning to do the same thing in September. We rented last trip (a couple of weeks ago) and decided to look into buying so called into a few shops to see what they had.
As others have said, the Chinese imported bikes are cheapest - I think about 180TL in Migros for a full suspension bike.
We also spent some time at the cycle shop near Migros (as described above) and the guy there speaks good English and appears to know what he's talking about. He sells Chinese, Turkish and even some German bikes - the latter are fairly pricey 500+ EUROS !. He also has a good selection of types - road, mountain, hybrid etc.
We are probably going to go for mid range alloy frame Turkish (Salcano I think) which are around 300TL each for a mountain bike - we need lightweight ones so I can easily carry them upstairs to the apartment!
I did also take a quick look in the shop in Calis up from the Dolphin roundabout on the way to the Sunday market. He doesn't have so many in the shop but again has Turkish bikes at around 250-300 TL.
Hope that helps - let us know what you end up with.
Ian