The primary problem with insurance (not just health) as a market is that people have very little idea of what the risks are. Since insurance is about protecting yourself against risks, not understanding the risks means that you may buy insurance that you don't need and not buy insurance that you do need.
In the United States, most health care insurance is actually purchased by employers. Since businesses can hire an actuary to examine their risk pools, this partially offsets the problem from their perspective. However, this makes the individual problem worse. Since more established, older people are more likely to get their insurance through their employer, they don't develop the wisdom to advise younger relatives or acquaintances.
The US system involves children getting insurance from their parents, young adults skipping insurance, and older adults getting insurance from their employers. People who buy their own insurance are often lower income people who use a lot of health care. They have not yet made it to the point of getting a good enough job to get employer-paid insurance, but they have immediate needs. And they may not know how to play the tax system as well as employers. Wages are taxed but benefits often are not. So employer insurance is untaxed while individual insurance is paid out of taxed wages.
Another problem is that the US system is disconnected. A doctor works for a practice where a nurse bills an insurance company which is paid by an employer who employs the patient who needed the care. That's four steps from doctor to patient in payment. But the doctor provides care to the patient directly. This leaves both rather insulated in terms of payment.
Several times the relationship goes the wrong way. The nurse who bills the insurance is an employee of the doctor's practice. She can't tell the doctor what to do. And the patient can't tell the employer to pay more money for insurance except in terms of salary negotiations--when we get back to the problem of people not understanding risk.
The employer bears the burden of costs, but it's the behavior of the employee/patient which incurs them. Bad diet, not enough exercise, live in a bad environment, not take recommended pills. All that causes health to worsen but they can't offset increased costs vs. behavior changes, only behavior changes vs. health. A market where cost doesn't matter isn't much of a market.
Usually a market is where you can either participate for benefits in exchange for money or not. But in health care, you choose an insurer and pay before you need the product. How do you evaluate that? Particularly when if you wait until your child is sick, you'll be willing to pay anything. But it's too late then. You've already chosen not to purchase the critical coverage or only have the bad doctors from which to choose or whatever.
A final problem is that medical care is too expensive. In Iowa, there is a single patient who costs $18 million a year. That's $820 for every enrollee before they can pay any money towards their own care. So if people are asked to pay for their own insurance, they consistently say it's too expensive.
When people buy insurance, they look at expensive but unlikely to use coverage and figure that they can do without it. Which works fine most of the time, but when the giant bill goes to you, that's your problem.
It's also worth noting that the Obamacare "fix" for this has been to make insurance more expensive, pushing deductibles and copays higher to keep the premiums low. So Obamacare plans cover less than the plans that they replaced. But this cuts down on the number of people involved in paying. For most people, the insurer doesn't pay anything. They pay for their care directly to the provider (or more likely the doctor's office manager). The provider just reports the amount to the insurer in case the deductible gets met that year.