I don't use Twitter but the weather channel app on my iPhone works without any problems after the latest update.
Have you tried restarting your iPhone to see if that helps? Press and hold both the power and home buttons until the Spple logo appears, then release both buttons and wait for the iPhone to restart.
If that doesn't work, there's the possibility that there were problems with the updates on your iPhone. Try deleting one of the problem apps and redid load it to see if it helps.