Hmmm…. a nice idea. The only issue is that you’d see tweets like
“This is the funniest thing EVER youtube.com”
or “I can’t believe the President did that news.bbc.co.uk”.
Which doens’t give as much context as the full URL.
Perhaps the solution is somewhere in the middle. Show only the first , say, 50 chars of the URL. Enough to get context, but not so much to break things if the URL is huge.