The US health system is very highly regulated. For what an unregulated market looks like, see the software business. Consumer costs for software have been largely driven to zero.
The software market is a very different market from the health care market. An unregulated software market may not quite be a free market due to monopolies and vendor lock-in, but for consumers, it's quite possible to say no to any software purchase. That's not the case with health care, where often your options are to pay or die. That means that in a profit-driven health care market, prices can be raised indiscriminately, especially in the case of the US where health care prices are often kept hidden, because the people who can afford to, will pay anyway. There's better margins in providing health care to the rich than in providing it to the poor. It's profitable to let the poor go without healthcare.
This makes the healthcare market a very poor market to let be government primarily by a profit motive. Of course health care companies in other systems still make a profit, but they are very limited in how far they can rip off their customers compared to the US.
It's only hard to see when you believe what business lobbyists tell you. It's actually easy to see when you look at the numbers.
The real lesson here is: don't believe everything people tell you. Look at the evidence first.
Now there are plenty of government projects that are woefully inefficient, but health insurance seems to be one that governments are actually good at. Maybe not great, but better than an unregulated market.