If you notice that whatever update you post on your Facebook timeline or Facebook page automatically appears as a tweet on your Twitter, then it's more likely
that you've linked your Facebook account to your Twitter account.
If URL that starts with fb.me appears in some of the tweets, it simply means that the connection was done at : http://www.facebook.com/twitter
So, if you want to stop Facebook from posting automatically to the Twitter account, go to the URL above, look for the connection and click on "Unlink from Twitter" as seen in the image below :
You can also link your Facebook page to Twitter right there too.
>> How I Post Tweets from Twitter To Facebook Fan Page and Timeline