To change the order of the links in the Links Module, you need to update them one at a time. Go to the link you want to move and click the pencil and paper icon to the left of the specific link.
Scroll to the bottom and find 'View Order:'. In this drop down you have the choice to move your link Before or After another link in the list, that list can be found in the drop down to the right of View Order.
Decide where you want the link to appear in the list by using the combination of the Before and After view order and the listed links.
Then click Update to apply the changes.