In my experience, within Europe it is a relatively widely held opinion that the EU (and its precursors) did indeed play a big role in supporting peace and prosperity on the continent. (Which does not at all negate the support of the US.)
But perhaps a clearer example than western European peace might be the Balkans. I believe that the prospect of entering the EU has played a non-trivial role in helping convince the Balkan states to finally bury the hatchet for real. (Similarly the prospect of entering the EU has been used to motivate modernizing the post-soviet European states to great effect.)
> The EU by contrast seems to be a source of constant turmoil.
I think the turmoil you speak of is generally on a completely different order of magnitude than violent conflict. (Of course one would be easily forgiven for miss-interpreting the severity of this turmoil given how Europeans do seem to so enjoy moaning and griping about their lots in life -- justified or not. In fact sometimes I feel like it is actually one of the more unifying traits in Europe cultures. Then again, maybe that's just humanity in general.)