Many recommend joinging SGK (Sosyal Guvenlik Kurumu), which is the government-approved insurance system for Turkey. Once you have lived in Turkey for over a year, you are eligible to make use of this system by paying a fixed premium.
FYI, the Turkish government does not require foreigners over 65 living in Turkey to have healthcare coverage. Some expats decide to just pay for all medical treatment as they go. But terminal diseases or major operations can drain your savings or burden your family.