Blown bulb.
Reseat the bulb, then test.
If fail, replace the bulb, then test.
If fail, send service on your headlamps.
Also, when you replace a bulb, spray a little electrical contact cleaner in the socket, then apply a light coat of dielectric grease to the bulb base. It'll help prevent future corrosion from moisture. Do not apply a bunch, just a very light coating is all it takes.
The switch is probably bad or you aren't turning the knob all the way off.
On the left turn signal lever, twist the end cap to turn lights from OFF - Parking - ON - AUTO
Check fuse for tail/parking/dash lights
If you browse this website you will be able to find what you need: http://www.jeep4x4center.com/mopar/jeep-grand-cherokee/backup-parking-sensor.htm
If you have factory installed fog lights (driving lights) you have to pull out the knob on the light stem towards the direction of the drivers side mirror
sounds like you have a shorted wire somewhere in the circuit. If your jeep has a trailer harness or has rust at the rear of the jeep, look for wires that may be grounding out. Could be easy or a nightmare but this is a good place to start.
bad ground, check wires to see if they are cut/torn/melted etd.
Check fuse for tail and parking lights Could be a bad dimming rheostat Could be a bad headlight switch
What are the wire color codes on the tail lights of a 03 jeep grand Cherokee?Read more: What_are_the_wire_color_codes_on_the_tail_lights_of_a_03_jeep_grand_Cherokee.
The cost depends on where you take the Jeep Cherokee to be fixed. There will be a cost for the lights and other supplies plus a charge for the labor.
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check your fuses