You can get lovely clean single bed rooms for AUS$20 each for 2 people sharing inc breakfast. Yes and that means no sharing showers or anything!!!! WHAT THE HELL WAS I DOING IN SULTANAHMET ALL THAT TIME!!!!! ahhhhhhh
Sultanahmet is a very popular area for tourists as there are the famous "The Blue Mosque" and "The Hagia Sophia" in that area and silk carpet shops and historical hotels as well. So as you appreciate, to spend a night might be a little expensive but you can always find a budget friendly hotels though! As somebody mentioned above, you can always find a lot of hotels at different choices and prices in Taksim,which is the center of the city and very close to the old Pera Street called Beyoğlu today.Actually Beyoğlu is a very interesting street to experience, I can say the heart of Taksim! Anyway, around Turkey, besides hotels you can also stay in " pansiyon",like renting a room in a house or let's say "a family stay" in a way. It is very popular in the Eagean coast and also in the southern part,especially in the touristic summer resorts.
If you are visiting a village or a small town, do not get surprised if you are invited to spend a night or have dinner by the town people and they never charge you for that. You must have heard about the hospitality of the Turkish people I guess!