NOTE: I do not main ADC. I play every position relatively often except for Mid. I am best at Support, then ADC and Jungle. That being said, I've played a lot of bot lane as both Support and ADC.
Not to toot my own horn, so to speak, (because I play a lot of support), but the Support always carries lane. That's why they're there. Supports are usually utility champions with good early scaling so that they don't need items as much. ADCs are generally not as good early. Those that are may be found in other lanes from time to time. (Ashe, Vayne, Graves, and Ezreal come to mind.)
Now, it is possible to survive lane and carry late if your support was bad. It's usually a lot harder, because you have to play really carefully in lane, and that limits the amount of farm you can safely get. I would say the easiest champions to lane/win from behind would be Jinx (if you land a few good rockets, you can turn the gold difference on its head), Ezreal (he has amazing range, and can farm from a distance), Sivir (her shield gives her amazing survivability, which means she can risk a little more if played well), and Ashe (her CC makes her very useful in teamfights, even when starved early.)
I usually play Jinx, and have found that if I play correctly, I can carry even if my support feeds regularly. Though a lot depends on what the enemy does with their advantage, you can definitely take steps to mitigate your loss.
~CrimsonCobra3