If your room does not have a safe box then why not ask or email the hotel and ask them if they have a safe at the front desk or in their office. All hotels in Turkey have a safe..or at least they all should.
Put your money in an envelope...seal it and then write your signature across the seal and then put it into the hotel's safe. Only take out enough cash as needed. Leave your other valuables there as well...passports, credit cards, bank or debit cards etc..
I also agree about using the ATM's. A better idea than travelling around with loads of cash.
One thing I would just like to point out is if you use a ATM please use one that is located outside of a national bank and also during regular business hours. Mon-Fri 9-5.
This way if you experience any problems with your card you can directly report it at the bank. Also do make sure that you advise your bank of your planned travel to Turkey. If not the bank after a transaction or two may freeze your card after one or two transactions.