In general, Facebook treats Bitly links as though the long URL had been shared directly. For example, if you create a short link to a Youtube video and then share that short link on Facebook, it should look the same as if you had shared the Youtube video directly.
If your links aren't displaying correctly on Facebook, the cause may come from the destination page. First, check to make sure there are no issues with your destination URL.
If there are no issues with the destination, use the Facebook Sharing Debugger to get more information about what's going wrong.