Visit the website of the Ministry of the Interior https://permissions.gov.ct.tr/staypermit and click on the "Şimdi kayıt ol / Sign Up Now".
Then, in the opened questionnaire, state your passport data, names (only first names) of parents, address of residence in the TRNC (one in one as in the lease/purchase-sale contract, if any), local phone number, and e-mail address, then press "Gönder / Submit".
The system will send an SMS to the specified number with the registration confirmation code. Enter it and proceed to fill in the data to obtain a residence permit.