Decided to upgrade to LED tail and turn signal light in the rear. Everything worked fine with the regular bulbs. I got an electronic flasher unit that is compatible with the LED's. Turn signals work fine but as soon as you turn on the tail lights the turn signal stop working. Any ideas on what the problem could be. The turn signal switch checks out per the shop manual.
  

are the LED's you installed able to do both though? I think some can and some can't.

The LED lights are the 6x2 ovals same size as the ones they replaced. Plug from the existing wiring plug right in to the LED lights.
Tow lights per side. Wired for separate turn signal lights on the outside oval with the inside oval having the tail and stop light function. Plug in on lights says tail, ground, stop. Everything worked normal before I plugged in the LED's. I read a lot on ACK's site about the conversion and I'm going to try a different flasher unit after I check some more grounds

Plug in on lights says tail, ground, stop

So where's the turn signal connected to?

Suzuki uses individual filaments for turn signal, stop & tail lights - unlike a number of "domestic" brands which share the stop & turn signal filaments - you say the LED lamps you have chosen have two lamps per side (presumably this is what you mean by tow) Suzuki uses three - this is most likely where your problem originates, and yes, I'm telling you the same thing as doughboy, just in a little more detail.

I have a custom bumper with two separate 2"x6" oval lights per side. Each light has a three pin plug labeled Tail, stop and ground. The ones with filament bulbs have one two filament bulb wired for separate stop and tail light. The LED lights have the same plug with with the same labeling. The inner two lights are wired for the tail and stop lights. They work equally well with both types of lights. The two outer lights are wired for turn signals. With the filament lights, it works great. When I put in the LED lights they work until you turn on the running lights. Then they stop working. I have checked all the switches and wiring and it checks out fine. As soon as I replace the two turn signal lights with the filament lights every thing works great again. The front bumper has a single 2x6 oval filament light on each side wired for running light and turn signal. The EP34 flasher work with the filament bulbs as does the stock flasher. The only thing I have left to try is wiring both the tail and stop sides of the plug on the LED light together for the turn signal. I'm hoping the extra current draw might fool the flasher into working.

I had the same problem go to parts store and get a electronic tail light coverter .it puts your stop and tail wire in one.