Bad News, You probably need a new Body Control Module (BCM) its located under your steering column on the inside. Where all the fuses are. That's the BCM. It holds all the really and whatnot for your cars electrical system. Fortunately look on the net places, there is a very cheap fix with a bit of DIY instead of shelling out $600 for a brand new BCM. Good Luck!
There is a way to fix the problem pull the entire bcm board apart the relay is ep2-31lt cut the top of the relay off. About 3mm up from the prong the plastic cap will come up and inside the relay is a couple of connection. in the middle if you look between the two coils they is a grey build up, if you take pieces of paper and clean off the build up the turn signals will work. I hot glued the cap back on and reassembled and everything works as it should. You have to remove the relay to do this it would be a good idea to have a soder gun and a soder sucker because this a 10 prong plug. I have worked on the tiburon that i had for 2 months and figured it out, don't pay the $800.00 for the new board. If you would like to ship your board i would consider fixing it.
If you fix the problem causing the light to come on, it will go off.
If you turn on your 1999 mustang v6 car and the right signal light stays on and doesn't blink and you switch the car off and then the left signal light stays on without blinking, then you need to see a mechanic.
i have an 04 and the same has happened to me my left blinker just stays on with the car on or off its your body control module, BCM, and one of your relays went out bad part is they are built into that BCM and a new one is around $950 call Hyundai of North America and put in a claim in enough of us do this they will recall the faulty BCM and give us a new up graded one the number is 1-800-633-5151 this has happened to alot of tiburon owners we need to fix the problem
One of the bulbs is burnt out.
You have a burned out bulb on the right side. Even a the side marker bulb will affect the turn signal
this could be because of a faulty body control module, BCM, they are about $950 from the dealer and all your relays are built into it for me my blinker just stays on all the time with the car on or off call Hyundai of North America file a claim if enough of us do this we will get our money back on the part and or get a new up dated one without the flaw the number is 1-800-633-5151
Most likely, one of the right side turn signal bulbs is burned out.
First thing is to check for a burnt out bulb in either the front or rear turn signal for the side that stays lit. Second, if the bulbs are fine then check for a bad turn signal fuse/cannister. Hope this helps.
It means that a light bulb is out on the side where the light solid (not blinking) it could be in the front or rear of the vehicle.
Sounds like you simply have A burned out bulb, Turn the signal on and look at all the lights to see if any are burned out. Could be A marker light if its in the turn signal circuit.
The turn signal flasher relay is defective, replace it.
Get a new blinker unit.