If you look at the vision correction and plastic surgery markets, you'll see they function quite well! Providers compete on a combination of quality and cost. So I don't think there is anything intrinsic to the free market that suggestion it's to blame for ever increasing prices.
My personal hypothesis is that Americans get a lot more healthcare than most countries provide, combined with higher prices. Not all of it is all that beneficial.
If you want to read something really eye-opening, check out this McKinsey report, bottom of page 14[1] It basically compares US healthcare spending across categories, adjusting for GDP (expecting the US' higher GDP means we spend more).
For inpatient care (hospital care), the US actually spends inline with what other countries spend. For long term and home care, the US spends less than other countries. Same thing with durable medical equipment.
Almost all the "excess spend" is in the outpatient setting. American's get a shit ton of procedures done that other countries just wouldn't do. Have a hernia? Providers in the US fix that, where other countries might say "we'll do something if it becomes a problem" (for example).
[1]https://www.mckinsey.com/~/media/McKinsey/Industries/Healthc...