I have flown Turkish Airlines, but not with firearms.
Agree with the suggestion that using a knowledgeable travel agent when flying with firearms internationally is the way to go.
On the Turkish Airlines website, there is a link to a pdf document that provides some instructions. To me, it looks like a nightmare of a process.
What to do when carrying a sporting rifle on flights
Transportation of hunting equipment on domestic flights
1. Go to police checkpoint and hand over your rifle.
2. Contact the counter and give rifle information.
3. Go to the Ticket Sales Office and complete rifle carrying payment (if there is a charge).
4. Contact the officers at the counter to obtain a rifle label.
5. Go to the police checkpoint and complete remaining procedures.
Transportation of hunting equipment on international flights
1. Go to police checkpoint and hand over your rifle and the necessary documents (sales invoice of the rifle, boarding pass, a photocopy of passport/ID).
2. Contact the counter and give rifle information.
3. Go to the Ticket Sales Office and complete rifle carrying payment (if there is a charge).
4. Contact the counter to obtain a rifle label.
5. Proceed to the police station, with an assigned security personnel to complete the remaining procedures.
6. With a security personnel, apply to the Customs Enforcement Office to complete customs procedures.