Many thanks. I see no reason why you cannot but as Turkey is not part of the EU it will not be possible to use the streamlined European claims process. Equally I cannot see that the UK courts would find sufficient basis to claim jurisdiction to hear the matter because although you live in the UK, both the dentist and the place the contract was performed was carried out in Turkey. Accordingly under the Rome Convention, unless you have a written agreement with the dentist that provides that the UK courts have jurisdiction under the contract which I imagine is unlikely, you will need to consider bringing a claim in the Turkish courts for which you will likely need to retain a Turkish lawyer or process agent unless you are familiar with the Turkish legal system.
There are now many "medical tourism" travel insurance products available through which you can specifically insure yourself against medical negligence received abroad. If you purchased a specialist policy before you travelled you may have a claim against that policy for an negligence that occured. In addition some travel insurance policies include a legal expenses cover provision which may be able to assist you with bringing a claim in Turkey.
Finally the dentist may provide some form of warranty insurance. Some foreign dentists carry such cover as a way of reassurance to foreign customers as to the quality of their work. I assume you will have already contacted the dentist but it may be worth researching his promotional material to see if such a warranty is offered and if so through which independent insurer.
Failing all this, you can find an English speaking Turkish lawyer on the following list who may be able to advise you the likely costs involved in bringing a claim, what you chances of success are and whether there are any insurance based funding arrangements (like no win no fee) available:
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