Welcome to our community and thank you for sharing your situation with us!
As you already mentioned, it is indeed impossible to get married in your home country, as she could lose her residence title in Austria if she returns.
If you are planning on getting married in Turkey, and your girlfriend has a passport that allows her to travel, then you need to take certain preparations. I'm not an expert on marriage, so please make sure you check all the legal regulations. Here is a website I found listing the process for "foreigners": http://www.allaboutturkey.com/marriage_foreigners.htm
Especially this paragraph is important:
"Foreigners (or local people) who want to marry in Turkey must provide their marriage licences issued by the proper Civil Status Register of his/her own country and proove that they're not already married at the moment. Otherwise, Turkish relevant authorities do not conduct these marriages. This means that the married couples cannot again be married in Turkey under the Turkish law."
Please have a look at previous cases from our platform, where more details on what to do and who to contact is discussed:
Marrying a refugee in Turkey - what to do first and can we travel afterwards?
I want to marry my gf she from Finland and I am illegal in Turkey? What can we do?
What's the best way for me (German) to get married to my Syrian boyfriend, who is staying in Turkey without a permit?
I hope this helps. Let us know if you have any more questions or need further assistance with anything.