If you stay up to 90 days in Türkiye, you do not need a residence permit (ikamet izni); for a longer stay it is required.
For the residence permit, Türkiye requires valid health insurance for the whole period of stay applied for; anyone who is GSS-insured in Türkiye counts as validly insured, and the Göç İdaresi (Presidency of Migration Management) names as proof the SGK certificate of benefits under a social security agreement, an SGK provision certificate, the certificate of a GSS application, or private health insurance.
Anyone who can receive health services in Türkiye under a bilateral social security agreement and can prove it needs neither private nor public health insurance for the residence permit.
For residence applications made inside the country, private health insurance from foreign companies is only recognised if it was taken out in Türkiye; for applications made from abroad, policies taken out abroad are recognised if they offer the minimum cover required in Türkiye. How people from Germany proceed in practice is something you clarify before booking with the Göç İdaresi or a Turkish consulate general.
For private insurance for the residence permit, the supervisory authority SEDDK sets minimum cover limits and determines that sudden illnesses may not be excluded from cover. The amounts change; you check the supervisory authority’s current version shortly before taking out a policy.