When you register with the Muhtar in your area he will give you a KONUTTA KALANLARAAIT KIMLIK BILDIRME BELGESI (FORM 5).
I have been speaking with a lady I know who has children she had to obtain residency for and the children do not have the same name as her due to a second marriage,this makes the process of going through the residency process a little more complicated.
You need
Full birth certificates with the mother and father's name.They have to be translated and notarised (this can be done in Turkey.)
Death certificate if one parent has died.Translated and notarised.
Divorce papers.Translated and notarised.
Any court order that determines custody of children or child.Translated and notarised.That is if the document exists.
Written permission,preferably with a sworn affadavit from a solicitor,from the other parent.Again translated and notarised.