Bizarre indeed! And this only affects IE9? (BTW, my IE9 is OK at bitly.com) Is Chrome OK?
Have you tried clearing your browser cache?
Is everything else about the page normal? I was going to suggest that perhaps an additional stylesheet (accessibility) is being applied... but just the word "Copy" in links - unlikely!
On that bitly.com page is the "Customize" link OK (next to the Copy link)?
Have you scanned for malware? But that sounds an unlikely cause, unless it is trying to be mega-subtle?!