This version of Twitter is fully working on iOS 4 & 5, and I even recommend it on iOS 6 if you hate how many tweet notifications you get on the newer versions.
https://www.dropbox.com/s/6klmoomxtd0ldr8/Twitter-v4.1.ipa?dl=0
https://www.dropbox.com/s/6klmoomxtd0ldr8/Twitter-v4.1.ipa?dl=0