Don't they still subsidize their agricultural sector, and dump stuff on the international markets and hurt non-EU farmers? I consider that to be very uncompetitive.
The common agricultural policy is largely an ecological and humanitarian disaster. However, it's only one aspect of the EU, and most of the member states would have similar subsidies if they were outside the EU anyway. Even the anti-EU parties in European nations tend to favour maintaining the subsidies (e.g. UKIP policy is to maintain the single farm payment after an EU exit). The US isn't in the EU and still has a significant program of agricultural subsidies.
The USA subsidises its agricultural sector, I agree that the EU does do it more. But then Japan puts in more direct subsidy to its agricultural sector than the whole of the EU. Still, whatever your view on subsidies, it is a much better thing for people to be complaining about than Europe being on fire. Again.
It's not nearly as bad as it was when payout was still strictly by production. Nowadays, farmers are paid to let fields fall fallow so there's no insane overproduction, hence less market crashing abroad.
Some sort of CAP is a necessity: The point is to keep local farmers in business as to keep local production capacity (not necessarily production) alive so that when shit hits the fan when it comes to imports, Europeans won't starve.
Yes, there should be further fixes. But it's much better than in the past, and I think the fishing policy should be dealt with first.