I think you will need to be in Turkey to open the account or to activate internet banking on an existing account. Whichever bank you choose, you will need to nominate a mobile telephone number in person for the "one use PIN" they send to your mobile to log on.
We banked with TEB, Garanti and Seker Bank when we lived in Fethiye. TEB & Sekers' internet banking is pretty straightforward and in English, as are most of the banks in Turkey. We did find Garanti Banks' internet site the easiest to use though.
I think if you nominate a UK mobile you should be able to operate the account from the UK, we never tried as we lived full time in Fethiye.