Closing time is a bit nebulous in Istanbul. I've never seen one posted, but I've been getting by assuming that they close when all the customers leave and/or extremely early in the A.M. Especially in the Beyoglu area or around the coastline, bars and clubs are open and really active until 3 or 4 in the morning (at least!), while dance clubs will stay open until 5 or 6.
If it's a monday night, however, and you are the only people in a bar and not drinking too much, you might expect a little pressure from the management to skeedaddle early, like maybe 1 or 2 a.m.