When your e-Visa expires, you cannot apply for a new e-Visa without departing from Turkey; but you are advised to apply at the Provincial Directorate of Migration Management for a residence permit.
Please click on the following link to get the more detailed information regarding the application for a residence permit and other types of visas (work visa, student visa, etc.)
http://www.goc.gov.tr/main/En_3.
If you overstay your visa, you might be asked to pay fines, be deported, or ban future travel to Turkey for a specific period. My child is included in my passport.