HSBC do offer Turkish mortgages to non-nationals, I have recently got one for a German client at the Bodrum branch. If you have an HSBC account in the UK, you would still need to open a new one in Turkey.
We offer mortgages in Turkey via our UK based mortgage advisor. He is Turkish and deals with all the major banks where in most cases he knows the manager personally, to get the best deals including off plan and re-mortgages which are now widely available. Mortgages are available of up to 85% (lending limit is based on monthly income) and a 'mortgage in principle' offer can be obtained within a couple of days. He is also happy to call anyone wanting financial advice. Fixed or tracker rates, interest only mortgages available.
It is possible to approach individual banks yourself, but in a country where you are unlikely to know the language and way of doing things, it is often advisable to go through a broker who can check all banks for the best rates & deals.
See
Turkish Mortgages