How are you retrieving and processing the feed you’re displaying on your website? Are you including [node:127] when you retrieve your timeline (setting the include_entities parameter to true)?
If you do so, you’ll notice an entities attribute in the status object, that contains parsed and structured data (url, media, hashtag…) extracted from each Tweet. The urls entity is the one you’ll be the most interested in. For example, with Tweet #93325601502543872:
“urls”: [
{
“expanded_url”: “http://symp.ws/n”,
“url”: “http://t.co/JRRz8YU”,
“indices”: [
78,
97
],
“display_url”: “symp.ws/n”
}
],
You can use display the expanded_url, but be sure to hyperlink it (href part) to the url.