Updated on March 6th, 2015..
The PDF doc in Turkish on Turkish Post Office website (www.ptt.gov.tr) says sending a postcard overseas (by plane with delivery confirmation) costs 5.25TL. Without the confirmation it costs 2.50TL per postcard.
Currency data courtesy coinmill.com
This is what I did:
I went to the nearest PTT, and got a number from the ticket machine, then started waiting. While looking on the posters for information on the cost of sending postcards, I saw an PTT employee who didn't look that busy to me (packaging, etc). I walked to him, and asked for the cost of postcard stamps. He showed me the stamp price listings for different countries, and also told me that he would stamp my cards right there. I guess he helped me before it was my turn because it didn't require any electronic transactions, and he was free.
If you are a foreigner, then I would definitely suggest asking around after you get a ticket number. Turkish people are very helpful to foreigners (especially when they try to say some Turkish words), and they will probably just get your stuff done right away. If you want to be prepared, then ask a Turkish friend to write down what you should tell the employees beforehand. I don't think all of the branches have employees with ESL education.
Here is the list of PTTs in the Beyoglu area. Unfortunately, the PTT website is not really user-friendly to foreigners. Just change the "Ilce secin" (pick a district) to be able to see the other listings.
Also, I found most of the PTT locations on Google Maps, however, it doesn't include every branch. It could be misleading.