When I bought my 99 TJ w/ tube fenders, the PO had used the side marker light along with a jumper to the turn signal wiring and basically stuck the normal side marker bulb in the front of the fender as a turn signal w/ no side marker light.
It flashed fast, and when you turned on the headlights, both turn signal indicators on the dash would illuminate like you are showing.
When I replaced this cluster f*** with a Gen Right Stealth marker light kit, the directions had me wire the grounds from the two LED's (side marker and turn) together and hook to the turn signal ground. Then the "blinking" wire from the original turn signal was wired to the LED turn signal and the the remaining wire from the original turn signal was wired to the positive wire of the LED side marker light.
The side marker lights were not used. I replaced the original flasher unit under the dash with a replacement flasher unit made for LED's that I bought from the auto parts store and everything worked perfectly.