I should think that most "high-street" bank branches would change foreign money into Turkish lira. If in doubt, you're most likely to find ones that do in the Taksim area and other "business" areas of Istanbul. That said, Turkey has a system of nation-wide banks that have branches all over the place – city centres, towns, suburbs – so when you find a branch that offers the service, it doesn't really matter where it is: the exchange rate will be the same at that branch as at any other. Some of the bigger banks are HBSC, İş Bank, Ak Bank, and Halk Bank, but there are lots of others.

I'm not sure that the saving over an ATM will be very great, though, unless you're exchanging significant sums of money.

As I said above, you're going to need Turkish lira (unless you're spending your time in Turkey in a complete tourist cocoon): you can't rely on US dollars or euros to conduct ordinary affairs, even as a routine tourist. And ordinary shops in Turkey don't "negotiate" prices – that's a tourist special in the grand bazaar (and it might be practised in some of the traditional hans), but for ordinary purposes shopping there is the same as anywhere else in Europe: the price is fixed and you either accept it or you don't.