You're doing a lot better than other beginners! At least you ended up in the road, going the correct direction, and not being afraid to get in the lanes with cars.

But like others said, I would have gotten in the lane with the cars (the right lane of the two turning lanes) and earlier than you did. When the light turns green you can maintain the same speed as the cars as you go thru the intersection. As you come out of it you can merge back into the bike lane if there is one.
But if traffic was crazy, preventing you from getting over early enough, I would have done exactly what you did if there was a bike lane/shoulder on the road you are turning on to. If not, I'd still do what you did but I'd head straight for the sidewalk after the turn at least until the traffic that was behind you goes by you. This way you are not pissing off cagers by getting in front of them at the light and staying in front of them.