HOW TO KEEP TWITCH FROM GETTING BLOWN UP

So I recently started playing twitch as an AD carry. In game I usually get ahead of my opponent, whether it's by some kills or a few dozen cs. However, after I obtain this lead, I find myself getting completely destroyed in the mid-late game, due to how squishy I am. It seems that it only takes a couple auto attacks or a few abilities to kill me. When I dont get targeted by my enemies, we almost always win team fights because of the insane damage output I create. So my question is, as Twitch, how do you survive the late game when everyone on their team is targeting you and can burst you down in a few attacks??