Can foreigners open bank accounts in Turkey without residency permits?

The answer is, it depends. Just to amplify on my experience:

1. You need not have a 'Turkish' bank account statement to show proof of funds. Your bank statement from your country of origin can be accepted. Turkish statement is nice, but not mandatory. What you have to show in the investigation (or discovery process) is your ability to sustain yourself here (in Turkey) financially.

2. Turkish banks ideally like to see a Turkish Resident Permit with more than 6-months on it when you apply for your bank account. If you've been offered a 6-month permit or the remaining time is 6-months or less left, they will most likely NOT open an account for you. There is no hard and fast rule. These are all internal policies of the banks.

Depending on how much money you are bringing into the country, the rules vary from bank to bank and literally from branch to branch (each locale having their own way of looking at things). Aksaray (a relatively poor neighborhood) has a much more stringent requirement to open a bank account than the same bank's branch in Etiler.

Your mileage may vary.