The problem is that you cannot have any sort of sensible health insurance on an unregulated market.
On an unregulated market, insurers would begin to drop people who get chronically sick, or land in risk groups.
Before you can say: "so just forbid dropping insured people" - that's what US initially did. But there was a simple loophole around that - you don't drop people per se. You just start a new insurance company, and offer a lower rate to the healthy people who want to sign in.
The healthy go to the new one. The sick stay with the old one. Old one's costs increase, they get pushed to the everyone who was left, to the level that is unsustainable, and either the sick leave too, or the company dissolves.
So you need more regulation preventing companies from only taking up healthy people. But you can't go overboard, because if a sick person can sign up at will, people will only seek insurance when they get sick.
And you end up with the clusterfuck that US is right now. Which they are exactly for the reason of trying to avoid government intervention and government-set prices for services.
And then you have medical companies and doctors, who invent a new treatment, and the insurance is obliged to cover that treatment. So it's complicated negotiations there too. Without government interference, the medical companies have no incentive to keep the costs down really.
On the other hand, you - as a patient - probably don't want the government to tell private medical companies how much to charge for what. That's a new kind of problem altogether.