Firstly you should try to obtain a 10 Amp DC circuit breaker from Radio Shack or similar if you do not already have one....blowing fuses can be pricey and a pain.
When does the fuse blow? As soon as you try using the turn signals or the backup switch or simply at random such as going over a bump?
This fuse 39 circuit does not go many places but you probably have one of the leads shorting to chassis or engine.
First pull out the defogger relay point "E"....remote chance of the relay having a dead short on its coil.
It is doubtful that the turn lights are faulty as the flasher unit manages to drive them without problems.
Disconnect the battery and use a Multi meter on OHMs to measure from points to ground/chassis for resistance..do not touch the leads with your fingers!
So looking at the wiring diagram page 8W-12-27...the Dark Blue/White wire coming off fuse 39 may be touching ground somewhere.
First disconnect the backup switch and measure the wire to ground/chassis with it in the backup position....should not go direct to ground but should go through the backup lamps! Good chance there is an issue with the tow package. Go to page 8W-21-5 to trace the wiring there.
Unplug the hazard switch completely, you will now have no hazard lights or turn signals but does fuse 39 blow?
Problem is most likely the backup switch shorting to ground on its way to the tow package circuitry or a problem with the back up lamp wiring itself..go to page 8W-21-5 to trace that area.
In case you do not have the wiring diagrams download the 2003 KJ Service Manual here..includes wiring diagrams:
Index of /manuals/Jeep/KJ
If you do not come right I will upload some of the wiring diagram pages and we can go through them together.