You didn't say who had given you a new official address, do you have any documentation?
If it's any help we rent and when we moved recently we had to complete, download and print the form from this website
http://www.fethiye.pol.tr/ click the box on the right hand side (ikamet Tezkeresi).
We took our new rental contract to the Muhtar who gave us a form C5 (cost 30 TL), then to the Nufus Office (ground floor in the Tapu office building in Fethiye) who gave us another form which shows your address and ID number - no cost. Take all these plus 2 passport size photo's, your RP and passport to the passport police, they give you another form to take back to the Nufus office for stamping, which they then want back. No cost.
It took about 2 weeks for everything to come back with your RP book updated. Note: when you get the first form from the Nufus office get some photocopies because the Passport police want the original and you don't get it back. The copies will be useful for changing your address with your banks etc.
If you are already registered with SGK your records are automatically updated when your RP is changed so you don't need to go and change your address with them.