From the description it sounds like you were at fault. You are supposed to be as far to the right as is practiceable, but you have to pay attention to lane directions as well. You should not have been in the right turn lane, you should have been in the second lane over.
I have an intersection like that I go through every day. Actually in my case the first lane goes onto a side street, the second goes onto the freeway, and the third goes onto the freeway OR continues on the same road, and the fourth only continues. I'll either be in the third lane over, or I'll split the second and third lanes if the light is red. But in that situation, I expect people in the third lane to make their turn without regard to me. When the light is red I'll pull up to the front and make eye contact with the driver and check their turn signal (altough most people don't signal... grrr). When the light changes I'll jump hard and get out into the lane in front of them. Most of the time they are getting on the freeway and they will slide over when I am out of their way, but if they really are going straight I will slide over to the right so that they can get by.
They important thing is that you have to be in front of the car, or behind it; if you are to its side then get behind it, and don't try to go around.