Fuses blow from 1 of 2 things. A short to ground, or a bad connection. Obviously a hard short to ground causes the amps to go up immediately, thus the fuse pops.
But if you have a bad connection, it may work for a bit, but once it heats up enough from the bad connection, it will pop the fuse too.
My guess is you have a short to ground, thus meaning the bulb is shorted somehow, or you have a wiring problem.
Isolate the hot lead to the turn signal in a wiring diagram. Find it at the bulb, and wherever it terminates. Remove the bulb FIRST!!! Otherwise you will always have continuity between hot and ground. Take your DMM, and see if you have continuity between the hot lead and any ground source, if you do, you need to find where the hot lead wire is shorted, or run a new wire.
I usually run a new wire, but it's your choice. Saves a lot of headaches, seriously. But if you can think of a few obvious locations, check them first.
Scott