Original Enterprise destroyed in Search for Spock
Enterprise-A decommissioned after The Undiscovered Country
Enterprise-B "2329 - Lost (presumed destroyed) after the crew reports contracting an unknown infection"
Enterprise-C destroyed at the Battle of Narendra III, the C was encountered by the crew of the Enterprise-D in the episode Yesterday's Enterprise
Enterprise-D destroyed in the movie Generations
Enterprise-E repaired after the damage suffered in the movie Nemesis
Most of that comes from Memory Alpha. So yeah, they really didn't destroy that many Enterprises. The Enterprise B was the one that Kirk was lost from at the beginning of Generations, and the Enterprise C was only featured in a time travel episode, so it's only the Original Enterprise and the Enterprise-D that count.