How did the United States prevent their ships from being sunk by German submarines?

The change in Naval tactics that eventually changed the tide of the Battle of the Atlantic was the use of Naval Escorted Supply Convoys. Navy Frigates and Destroyers were also equipped with Radar, allowing them to detect a surfaced submarine at night.

This tactic, and the enormous shipbuilding capability of the U.S. meant that the U.S. could produce and protect more ships than the Kriegsmarine could sink. Merchant Aircraft Carriers were also employed, as well as the introduction of long-range bombers. Merchant AC's could carry up to 4 aircraft for dealing with any submarine threat.

It is important to note that the submarines during both wars were not true submersibles; they were surface vessels with a limited submergence capability. Today's boats can go much faster underwater than they can on the surface - the opposite was true in those days. For this reason, many attacks at night were made on the surface, and Radar removed that advantage.

The combination of Air Support, fast Navy Frigates and Destroyers with the ability to escort convoys all the way to England, and the shipbuilding capacity of the U.S. (in particular the Liberty ships) is what ultimately spelled defeat for the German U-Boats.