In English we use 'am' and 'pm' to signify if a time is in the morning or afternoon.......how do we do this in Türkçe???

Yes in written language, on tickets, on TV magazines etc. the 24 hour based system is used so you will see for instance 20:30. I don't know many European countries but in Germany they also use this system.

On the other hand daily life the same usage is not so common though it is acceptable as it sounds more formal.
If I were to use it in daily life I would say "öğlen ikide", "akşam sekiz buçukta", "sabah dokuz on beşte" etc.