Left tail light won't burn

There are a few possible reasons why your turn signal light is not working while your brake and tail lights are functioning properly:

1. Blown fuse: The turn signal lights are connected to a different fuse than the brake and tail lights. The fuse for the turn signals may have blown and needs to be replaced.
2. Turn signal switch: The turn signal switch, located on the steering column, could be malfunctioning and needs to be replaced.
3. Wiring issue: A wiring issue, such as a broken wire or loose connection, could be preventing power from reaching the turn signal lights.
4. Flasher relay: The flasher relay is responsible for controlling the turn signal lights. A faulty flasher relay could be preventing the turn signals from working.
5. Signal/light module: Some cars have a signal/light module that controls the turn signals, this could be faulty and need to be replaced.

Power flows from the 12 volt battery through Fuse 19 (20A) and to the Tail Light Relay. This fuse and relay are housed in the under-hood fuse/relay box located at the left rear of the engine compartment. When the Tail Light Relay is commanded ON by the MICU, power flows out from the relay to Fuse 27 (10A) housed in the under-dash fuse/relay box located at the lower left end of the dashboard. From here, power flows to the side marker lights, license plate lights and tail lights. If a fuse is blown, check for water in the tail light assemblies. If a fuse isn't blown, it could be the the Tail Light Relay has failed or is not being commanded ON by the MICU. The MICU wouldn't send a command to turn ON without first seeing the request from the headlight/combination switch, so there could potentially be an issue there as well. You can read more here: https://www.yourmechanic.com/article/symptoms-of-a-bad-or-failing-combination-switch-assembly

Check the fuses

If it is only the left, check the bulbs first, then fuses, then bad turn signal switch.

check for power at the wires going intop the tail lights, real common that the circuit board is bad, but unlikely both at the same time. The tag, tail lights and brakes are 2 different systems. Do the turn signals work normally? if so the wiring from the turn signal switch to the brake lights is GOOD. the stop light switch had\s a ORANGE wire that should have 12V if not ck STOP LPS fuse, when brake pedal is depressed, the WHITE wire is 12V (at the stop light switch) if that is OK, the turn signal switch COULD be bad. The wiring harness plug is near the back left under the tailgate, see if that is disconnected?

If it happened the way you described it. I don't see how it could possibly have damaged anything. I guess what you are saying is that the left tail lights are not working? ("4 way exterior dim lights"???) Touching the tail light wire to the turn signal wire will only cause your turn signals to light up. (I am assuming that the tail light switch was on at the time)
I think it is more likely that the brown wire touched ground. This would make a bright flash, blow a fuse and cause the symptom you are describing...I think you need to check the fuse again.
There are separate fuses for the left and right sides. The left side tail lamp fuse is fuse #7 in the junction block it is a 10 Amp fuse. The junction block is located on the right cowl side inner panel - We are NOT talking about the Power Distribution Center under the hood.

All of the above is also assuming that we are talking about a plain Cherokee, Cherokee SE, Cherokee classic or Cherokee Limited.

If we are talking about any of the Grand Cherokees, you may have a fried lamp outage module.

you need to check the under hood fuse block on the right hand side you may find that one of the red feed wires has no power from the box either a maxi fuse or the box it self you could also check for power with a test light moving back from the components to the main feeds under the hood

In the fuse box under the hood, fuse in position 20 is a 7.5amp fuse for the trailer left turn circuit, fuse in position 22 is a 7.5 amp fuse for the right turn circuit, if the fuses are good and you have power at them it is going to be wiring, at the back of the truck close to the driver side at the bumper is a gray connector that will take factory wiring to you trailer wiring the yellow wire is left turn/brake light, the green wire is right turn/brake light, and the brown is tail lights, and the orange wire is 12volt feed for if there is a battery on the trailer that needs to be kept charged for break away safty brakes.

Before going ahead and spending money on a new module, test it first. The module is located under the steering column. You'll find a Brown wire on one of the connectors (i just looked at a wire diagram). Do no disconnect anything. Back probe the brown wire, which is the power feed to your park lights. With the lights off, there should be 0 volts on you voltmeter. Turn the park lamps on, There should be 12V. If there is not 12V, disconnect the connector and try to check voltage at the pin that feeds to the brown wire, if still not 12V, replace the module. If you do get 12V, then module is ok, find where the 12v is being dropped before making it to the lamps.

Rear lights are double filament bulbs one side for the lights other side for the brake/turn signals so it still may be a bulb. Do u have a test light or volt meter you can check to make sure u have power there when brakes or turn signals are activated. Or just try another bulb.