Ten days for about $600 one-way Izmir to Antalya is a reasonable price. I know of less expensive rentals but there is a very substantial return fee.
The "cheapest cars" in my experience are however tiny, extremely under-powered commuter cars. They are quick, nimble and economic in town but not very good on the highway--especially in mountains or with a full load or for a long drive.
You can always try the smaller local car rental firms, but many are likely to refuse one-way rentals and again you will almost certainly pay a hefty return fee.
I would expect around $1000 to rent a Renalt Megane Diesel or similar for 10 days including return fee from the company I use. They have offices in Izmir and Antalya so perhaps the return fee won't be as high as I imagine. If you want to try, go to www.akustik.tc
My agent, Gözde can be reached by email at firstname.lastname@example.org
She is in Bodrum, but can arrange for rentals from other offices.