Initially, you need to apply for a residence permit. It should be both the parents and the child. On the basis of a residence permit, Your child has the full right to study at a Turkish school. This applies to both public and private schools. Moreover, the state education is free for foreign children, as well as for local ones. However, if You want to send your child to a free school, then only the school that is assigned to Your address can accept documents from you.
Then, after You have decided on the school, you need to write an application to the Director with a request for enrollment. The application procedure is exactly the same in both public and private.
Next, you must provide a certificate and personal file of the child, certified in advance by the Ministry of education of Turkey. Of course, if he has already studied somewhere before.
In private schools, an additional contract is drawn up, which specifies the dates of payment for tuition. Most private schools may require you to make a payment a year in advance.
Both private and public schools in Turkey are taught in Turkish.
There are also Russian private schools, but they are all paid.