I am a female, age 65, and was in Istanbul for three days (solo) just after the election in November. I had the opportunity to stop for a few days as a layover after a tour in Uzbekistan (an amazing place, by the way). I had been in Istanbul with a Rick Steves tour a year ago, so felt confident that I could navigate the city on my own. I'm very glad that I did this, in spite of the state department's warnings.
I found everyone friendly and welcoming. There were plenty of people out and about on the streets, tourists and locals alike. I met people from all over the world (but no Americans). The hotel was lovely and its staff especially helpful. I shopped at the Spice Market and the Grand Bazaar, where the merchants invited me for tea while we discussed my purchases. I took the tram to get around with no problems.
I did notice, since I visited twice within a year, that security was tighter. There were many more police on the streets and near tourist attractions with serious expressions and serious guns. The airport has more security than ever. One suggestion - give yourself LOTS of time when you go to the airport for your return flight. I went through no fewer than six security checks before boarding my flight, so you don't want to be rushed.
I wouldn't bother trying to hide your American citizenship. Somehow, they can pick you out anyway. I was alone, dressed modestly, and barely opened my mouth (to speak English), and they still seemed to know that I was from America. People were interested in a friendly way about where I was from, what I thought of the election and so on. It turned out that one merchant had a brother who lived very near me in the States, so we talked about his hopes to visit. Another merchant asked, "Were you afraid to come here?" I said I had decided not to be - the only attitude you can have and not drive yourself crazy. Bad things can happen anytime and anywhere. I happen to live in Minneapolis and drove over the interstate bridge that collapsed into the Mississippi River just one week before that happened.
If you're a bit nervous about your visit, I would suggest you hire a private guide for at least part of your time in Istanbul. A good, knowledgable guide can help you relax and have a good time.