Why does my turn signal/ hazard light fuse keep blowing out

Is the trailer wiring new or was it already fitted and the problem has just started happening? Does this problem happen all the time or only when the trailer wires are plugged in? This may sound silly but does the weather affect the problem? For example does it happen after it has been raining and stops when the weather dry up? Have you tested the wiring with a voltmeter/multimeter for actual live voltage?
Using the multimeter on DC volts start at the actual light fitting and work your way backwards. Remove the bulb and test for voltage in the hole the bulb fits into. Next test behind the bulb where the wiring connects to the light fitting. This tests the light fitting itself. Follow the wires back and find where the trailer wiring connects into the wiring harness then check for voltage before and after the connection. This tests for issues between that connection and the trucks rear lights. As you follow the wire look and feel for damage as a small cut or break in the wire can cause big issues.
Check any Ground (earthed) points aren't loose or corroded. If there is still no voltage the next place is the fuse box. A lot ot tail light issues are often in the wiring somewhere near the actual light fittings.

Electrical issues are always a bit difficult to diagnose. Since I have not heard the wiring harness to be an issue on these, I would first look into if there was some add on causing an issue. Did you or someone add a trailer harness? Some type of additional lighting (like an LED bar)? Even possible a radio or under dash work? If not then do a once over and look at where the wiring comes through the firewall, or through the body to feed the turn signals. You just have to systematically follow the wiring from the lights back to the switch until you find the culprit. Let me know what you find and I can try to assist further.

you obviously have a short to ground on one of the wires to the back (if it was the front the brake shouldn't blow the fuse). the most common problem is usually in a trailer light harness. if you look under the drivers side rear of the truck, down under the tail light there will be a plugin junction. pull this plug-in apart and see if the fuse still blows. if it does then the problem is forward of that connector, if it doesn't then the problem is behind that plugin. be sure to check for "green death" (electrical corrosion) at all joints especially any trailer hook ups.

May want to check the bulbs on the trailer to ensure the connections are solid and the wiring is in tack. May be a weak ground wire connection.

If the FUSE blows then it is confirmed that the circuit is SHORTED and so the heavy current drain make the fuse melt off.
So it is important that the circuit is traced to find the short. So remove the bulbs of the light circuits and check the holders, check the wiring harness, see for damage on the sections where the harness passes through. Now without the bulbs replace the fuse. If the fuse blows then the harness can be short to this circuit sections.

YOU COULD BE USING THE WRONG BULB.

check 4 trailer harness short lite are different elements different flashers

Do you have a trailer wiring harness? If you do, recheck the wiring there. If that doesn't help and you have properly checked the flashers then you need to check each bulb

Has this ever had a trailer hitch? If so follow the trailer wiring back to the mess, and the problem ... If not that ... I would check for a pinched harness on or near the steering column ... I would have said the signal switch too. You may have to follow this harness all the way back or foward ... it won't be fun.

Trailer? Am I to assume that you have a trailer harness? If so, that may be where your problem lies. Disconnect the trailer harness and plug everything else back together.