Yes, there are gypsies in Istanbul. As an American, I find that I cannot recognize them, but my Turkish friends say that they are very easy to spot. They claim that they wear many different clashing patterns of fabric, especially the women, and that they are in general shorter and darker than most turks. I can't confirm that's true, because there seems to be a lot of misinformation about gyspies in Europe in general.
If you want to go to a well-known Gypsie Neighborhood, try heading down the hill from Taksim towards the golden horn. There's an ancient church that is really visible and a district that sells only car rims and mannequins (I'm not making this up). I've had this area pointed out to me many times as where the gypsies live. It's a bit dicey, so please don't go alone!