The health systems in Europe are not radically different from the US system. The efficiencies of each are difficult to quantify without context. For example, the US invents most of the drugs and medical technology used by the world. Would this still happen if there was more invasive regulation? We can't know.
Besides all of this, think of every other area of the market where the government insinuates itself. Are public schools better than private schools? Almost never. Is the US postal system better than FedEx? Of course not. The government is not a commercial entity. The incentives and influences on it are not conducive to producing quality products at good prices.
> Are public schools better than private schools?
Yes, they are far better than private school, for those who can't afford private schools.
> Is the US postal system better than FedEx?
Yes, it's far better than FedEx for those who live in place FedEx wouldn't and couldn't operate profitably because the postal system's aim isn't efficiency, it's coverage to "everyone" everywhere, something no market solution will offer because that's not a profitable mission.
So it sounds to me like you don't understand the purpose of government and don't understand why government is necessary in such situations, because you can't think outside of your own self and realize that markets don't serve everyone.