Here's a great example from Elisabeth Rosenthal of the kind of scary price problems in American health care. It costs more to have a conventional delivery in the United States than it costs to have a cesarean in France or Switzerland or the Netherlands.
And note that this graphic is very careful to look at the actual all-in final amount of money paid. This is not a question of how the cost is allocated between the patient, the insurance company, the government, and the patient's employer. It's a question of how much money gets handed over from the people who pay for health care (patients, insurance companies, employers, governments) to the people who perform health care services. And Americans hand over a lot.
Perhaps someone will make the argument that America is gaining some important quality advantage over Swiss and Dutch childbirths in exchange for our money. But if you accept that we aren't, note that there isn't anything mysterious about the reason prices are so much higher in the United States than elsewhere. What other countries do is they write laws capping the price of health care services. The economic justification is that in all countries the purchase of health care services is heavily subsidized (because one of the best things a society can do with its material prosperity is ensure that sick people get better) so you need to use regulation to ensure that the incidence of the subsidy falls mostly on patients rather than on health care providers. In America, health care prices are largely uncapped so they get very high.