If you are paying by credit card, there's really nothing to worry about except to be OK with paying the USD equivalent of whatever EUR rate that is being advertised. You won't need to physically convert anything (use a card that has no foreign exchange fees). As for demanding to pay in cash in Lira, you can certainly do that but you'll be paying whatever Lira equivalent is of the EUR price, in the case of hotel rooms. Another way of saying this is, you won't ever pay what a local pays because you'll be treated as a tourist. I can't go on the US version of booking.com and not be quoted rates in either USD or EUR, so they know you're a tourist and there's no escaping that. Having said all that, just worry about picking hotel rooms and the like that are affordable to you (do the EUR conversion to USD using yahoo currency exchange or whatever). Turkey is really not expensive so it's not worth worrying about getting swindled - like everywhere else, you're a tourist and you'll pay tourist rates (but you can still get value for you money without feeling ripped off). Hopefully this makes sense.
When you're at the ATM, withdraw money in Turkish Liras only. Everywhere you go (tram, local shops, restaurants, etc), you'll be paying for goods in Lira and the prices will be written in Lira (unless you're at super touristy places like the Grand Bazaar). Prices that are marked in Euros will only be for hotels or heavily touristy shops or very expensive goods. It's supposed to be for the "convenience" of the shopper, but yes, it gives a lot of power to manipulate the exchange rates when they're set in Euros or other currency.