Kike most people have said don't be scared. I went to Istanbul on my own a few years ago and had no trouble.
Yes I was approached by people, mainly wanting to sell me carpets or jewellary or stuff and I did get stared out quite a lot being a woman on my own but I never felt in any danger. If someone stared I just looked away, if someone became too intent on persuading me to look at their shop I just said I was very sorry but I was on my way to meet friends. Like KARL said the attention can be a bit like a version of wolf-whistles in this country - just ignore it. If you do find it becoming a bit unwarrented (one guy did try to slide his hands on to my bum while selling me a bag) a loud shout of "Stop that!" works. Potential embarrasment is a good weapon.
As long as you remain friendly and polite, take the usual precautions you would take at home (don't go down dark alleys, don't 'advertise' your money/phone/camera) and be sensible you will be fine.
Turkish people are charming and friendly and I had wonderful conversations with them ranging from football to politics.
Have a great time and enjoy being free, single, able to do what you want and not have to put up with your scumbag ex anymore!