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The UNHCR has a resettlement programme that addresses acknowledged refugees that fulfil at least one of the following criteria:
1. Legal and physical protection needs
2. Survivors of violence and torture
3. Medical needs
4. At-risk women
5. Family reunification
6. Children and adolescents
7. Elderly refugees
8. Lack of local integration prospect
9. Family members in a third country
Germany, however, applies additional criteria on top of that. If you want to be considered for resettlement in Germany you need to prove in some way that you have a high likelihood of integrating well into German society and economy (based on age, education, language skills etc.).
So it might be quite difficult to be considered for resettlement in Germany, but you can still enroll in the programme and try to go to another European country. Maybe your sexual orientation is sufficient to fulfil criteria number 1 and/or 8.
If you provide UNHCR with compelling proof that there is no long-term perspective for you in Turkey they can search for a country that’s willing to take you in. Of course they will ask for your consent fist.
Or did you already try that?
In addition to the general UNHCR resettlement programme, Germany and its federal states established their own programmes in the past. But these programmes only address/ed Syrian refugees and/or have been stopped already. I don’t think there is other ways of legal migration to Germany apart from that (e.g. through organizations).