No, the tax free option is only available to foreigners working in Turkey, and even then is not worth the hassle, you have to deposit a large amount of money in an interest free account for the time you have the car.
Just to add to Abi's post, the car is virtually unsaleable when you have finished with it as it can only be sold to another foreigner and the same amount of tax has to be deposited by the new buyer in the interest-free account. There are, or were, many of them advertised on web sites. Definitely not a good option.
Best to bite the bullet and buy a car in Turkey, yes they are expensive, but they hold their value and you will get a lot for it when you sell it (subject to wear and tear). The thing to remember is that road tax is huge and increases with 'newness' and engine size. Anything over 2L is very expensive to tax.
Welcome to TLF & welcome to Kalkan from a fellow resident here.
Please PM me if you need any info. We moved here a few months ago.
I looked at the tax free car. It's not as complicated as some say - if you buy from a dealer they do all the paperwork, but you do have to have all your 'beans in a row' to get it.
As others have said, you also have to deposit a large amount in the bank and get the bank to gaurentee you against tax evasion etc - the import tax and thus deposit can be more than the car actually costs!
Then, and again others have said this, you will likely experience difficulties selling it.
In the end I went for a second hand 'Turkish'. I pay quite a bit in tax (but mine is a 3.2ltr Touareg - to cope with our Kalkan hills) - but still less cost than UK and mine has the LPG conversion - so cheaper than a flipping Duster to run.
Good luck with everything and enjoy your life in the sun.
PS: Dont get confused with my other response on an earlier thread about keeping a UK vehicle here in Turkey. I drove over in my campervan and am keeping that here for 2 years so that I can enjoy some travelling - but at 7.5mtr its not exactly a daily runaround.