Who has the right of way?

Im turning right into the road and the other person is turning left onto the same road but is in a dedicated turn lane. Who has the right of way
There are no stop lights

Other answer:

General rules say right turn before left, but in reality no one automatically has right of way unless someone yields it to him.

To quote the California drivers handbook:

When you turn left, give the right-of-way to all vehicles approaching you that are close enough to be dangerous. Also, look for motorcyclists, bicyclists, and pedestrians. On divided highways, or highways with several lanes, watch for vehicles coming in any lane you must cross. Turn left only when it is safe

Consult your local traffic laws – they vary from state to state and country to country.
Here, you have to turn into the lane closest to you, so as most roads with dedicated turn lanes also have two lanes in each direction, both could go at the same time. In practice, to save your paintwork, you have to assume that the other driver may not know the rules or isn't paying attention and be prepared to give way yourself.
You say there was no light. So what was there? There had to be SOMETHING. If there was a stop sign, it's the person who got there first. If you got there at the same time, it's whomever is on the right (or in real life, whoever is most aggressive).
So, if you think you have the "right of way" you're just going to plough through another vehicle. Common sense should be used. If in doubt yield. So you have to wait 10 seconds? Big deal.
Dan B:
In the absence of any other traffic control devices, signs or signals, you would have the right of way. The vehicle crossing opposing lanes of traffic would yield.
It is hard to imagine what you are describing. If you have a yield sign in your dedicated turning lane, you have to yield to any traffic. If you don't have a yield sign, I would think you have right of way. Make sure you use your signal light, regardless of being in a designated right turn lane. You want to make sure people know what you are doing.
don r:
People don't make signals and they get hit in these situations. The first one in the intersection has the right of way this time. Neither one of you used a signal, did you?
You have the right of way, but to the other person they may not care.
In general, you should give way to the opposite side that you drive on.
So in countries where you drive on the left, you should give way to oncoming cars to your right, and vice versa.
Therefore, the answer to your question really depends on the road rules in your country/state.
R K:
neither has a right of way. your supposed to complete your turn in the same lane you came out of. if your turning right, your supposed to complete your turn in the right lane. if your making a left, you need to be in the left lane when you complete your turn.