In the US, your employer will buy you health insurance, which covers some but not all of your healthcare costs.
If you don't need much medical care, your costs in the US will be small or nothing, essentially the same as in Europe, probably with much more expensive and luxurious service.
But if you do need extensive care, then even when a good insurance policy, you may be facing substantial costs.
A friend who moved to New York with a big tech company told me (lightly paraphrased to anonymise):
> I’ve had a surgery, a ton of dental work, and a new child. It’s cost me tens of thousands of dollars, but the experience has been consistently better than in the UK.
If that child had been ill and needed special care when it was born, that could have been hundreds of thousands. Whereas in Europe the cost is still zero, or only a little more.
There's also the issue that the insurance is tied to your job, and it's extremely hard to get decent insurance directly from an insurer. I'm not sure if this is still true in the Obamacare era, but I've read of extremely rich people being unable to get decent insurance.