For reference, I worked in a large medical billing office for years.
It's not a free market, it's the opposite. Completely socialized, just in a very round about way.
Medicare pays roughly 10% of any bill that's charged. It's a flat rate now (but remains about 10% of what insured people pay). Essentially anyone with insurance is "taxed" by having to pay 100% (then negotiated down by the insurance companies). You legally cannot not have insurance, if you make over a threshold (in part due to Obamacare), which has made the issue worse.
The above combined with the inability to see prices beforehand and the way the U.S. foots the bill for the worlds medical research has made it pretty insane.
IMO the ideal solution is to make insurance illegal. Hospitals then need to manage the cost and risk of running procedures. Yes, this means some people get worse treatment, however -- they also only get paid if you live and are happy. The incentives are then aligned, and the hospital manages the risk so better hospitals stay in business longer.
There may be some middle ground there, but having worked in the industry, it's the only "good" solution I see.