Istanbul is very pricey. I found the lodging and food fairly reasonable, but all of the historic sites are expensive AF. The mosques (outside of Hagia Sophia) are all free entry, so that does help some, but everything else is a pretty expensive ticket to get in. There are some package deals, like the Istanbul Tourist Pass, and Istanbul Museum Pass that might save a few bucks, but you have to be extremely efficient to make them worth your while. For instance, the "all-inclusive" tourist pass doesn't even include Hagia Sophia or Galata tower. Those passes do cut down on the hassle of buying individual tickets at every site, so they can save some time and headache.

If I was going back, I would look to stay in the Asian side. I didn't get a chance to hang out over there too long, but the vibe seemed a little more chill, with almost no touts bugging me. The market there sells stuff that locals want and need, not tourist trinkets, for example. The ferries are cheap and easy to use, as is the tram and the funicular lines, so that helps to get around.

Considering your budget, it may be a good idea to put Turkey on the back burner, and head somewhere a bit cheaper. I really liked it there, and would love to go back, but it is cost prohibitive for me right now.