Keep in mind Twitch's ultimate gives me alot of extra range. Plus it is unlikely you'll survive long enough against a a team + twitch's gattling gun speed shots piercing everyone. If twitch is getting killed during teamfights when he pops his ulti, he's either positioning himself badly or the team is derp tier.
Also, there's also an important detail about twitch people seem to forget. Twitch will usualy have at least +20% lifesteal and at least 30% crits by the time his build is mostly done. Now twitch's ultimate lets him hit several champs at the same time, and with his with infinity edge he should be critting around 500~1000 damage depending on how fully equiped he is. Lifesteal works for every champion he hits, and he's hitting multiple champs with every strike.
Even if someone targets him during the shootout, if they can't do some large burst damage against him in a few seconds, they either flash/ghost/run away or die horribly. The trick of winning against twitch is not letting him farm alot and absolutely not letting him get fed, or else he'll rape you over unless everyone else in his team is derp tier.
Twitch defines carries ever better then other carries. He's really weak on the start but if he gets enough time to build himself he becomes stupid powerful. Yesterday we had a jungler so we had 2 solo lanes, and I was on top against miss fortune and master yi. I towerhugged for a while and picked yi and miss fortune off everytime one of them went to base. After some kills I rushed infinity edge and it was already over by then. They surrended after I got a few rounds of double-triple kills.