Some Twitter users are unable to send links in Direct Messages (DMs) right now. Twitter is aware of the problem, but until it rolls out a fix, you can use a URL shortening service to bypass the restriction.
As CNET notes, all you need to do is take the link you want to share and first shorten it using goo.gl. Copy the shortened link and send it via DM like you normally would. It should work just as before.