There are threads about this issue on the MacRumors forums and the Apple Support Communities, with most people complaining that the problem started around the beginning of November.
Aside from reloading the t.co URL several times, removing the "https" from the URL, or switching to a different browser, there's no way to fix the link loading issue. Clearing Safari's cached files also helps, but only temporarily. Chrome and Firefox aren't affected, nor are links that do not use HTTPS.
In a recent post on the issue, The Verge uncovered a January 14 tweet from Timothy Hatcher, a WebKit Developer Experience Manager at Apple. Hatcher says the problem is lower level than WebKit (Safari is built on WebKit) and that Apple has a fix identified.
Hatcher did not specify when the fix might be implemented, and it is unclear if it could come in a standalone Safari update or if a full OS X update will be required.