American Express Disputes & Vayama
It seems your only chance of a claim would be to speak to Turkish airlines as if you booked the identical flights it would have flagged up in their system so they may be kind enough to refund you on the cheaper tickets paid or give you some kind of credit.
As far as Amex or your travel agent are concerned they had both acted in good faith by honouring your reservations and the fact that Amex approved the transaction shows they value you as a creditworthy cardholder.
In respect of your agent knowing you had made a booking unless you advised them they have no way of knowing you made another reservation through a different agent.