It is occasionally really difficult to obtain tickets from TA while in the US. People do manage it successfully, so I would continue to attempt to do it online. TA has a 1-800 number. Have you tried to call them?

I had trouble booking a ticket in March because I was attempting to use a Bank of America credit card, and they would not accept it because of some issues regarding fraud. I used a different card and had no difficulty.

When I have difficulties with Turkish Air that I cannot solve I call someone in Turkey and ask for help. Usually the people at my hotel are happy to do this for me. I would suggest that you contact your hotel in Bodrum and see if they can make appropriate arrangements. Failing that I would write to Rina at Insight Travels, explain you difficulty and see if she can help you secure tickets.