why do taillights go off when headlights are turned on with tow hitch on 2008 jeep

either you have two bad bulbs that or a bad ground (high resistance) and can not handle both circuits ...//is the trail hitch aftermarket you might want to check to see how its spliced in ...// if they did a bad job and did not solder or heat shrink the joints they are open to air and moisture and corrosion may be the issue ...I would check the bulbs first and power and grounds at the sockets with a test light or DVOM ...the bulbs should be 3157

He is correct it is a ground problem..look where the wires flex.most likely around the tailgate or in the wire bundle that is exposed when you open the tailgate. one single {most times black wire} is breached. Happen to me more than once. Easy fix......but you have to find it first.When you do....you will laugh at its only one little break