I think the problem is that all three sectors that you cite, health care, higher education and prison system, there is no free market but they are oligopols where only a small number of big players can lever their weight and keep others out.
"there is no free market but they are oligopols"
There used to be 12 private prison companies in 2000 but it appears there are only 3 now.
The start of the journey was privatization of prisons. This is where that path has currently led. It is what it is, ergo free market didn't work?
There are, in my opinion, certain aspects of government, which never should be outsourced for private gain. A prison company being one of the most extreme examples.
If other companies can satisfy the needs of the state, why not prisons? You wouldn't mandate that governments should build their own cars, for example?
In my opinion (and we can certainly debate this point) this should never be outsourced to third party entities. A rent-a-cop in a mall should not have the power to arrest customers (or even search their belongings) without calling the cops.
Outsourcing such functions have the inherent risk that the privat company cuts costs and corners and moreover hires unqualified and cheap staff, which makes a bad situation even worse.
Incarceration of a human being should never, ever be a license to aply additional torture in any way, shape or form. Private law enforcment with minimal oversight is a direct route to such excesses, which should not be tolerated by a civilized society.
You wouldn't mandate that governments should build their own cars, for example?
That's a rather silly analogy, don't you think? Consumer products usually don't violently intrude into the freedom and bodily integrity of other people.
[1] https://en.wikipedia.org/wiki/Monopoly_on_violence
edit : added link
But we choose a free market or whatever because we believe it will produce the best outcome. If it doesn't work as envisioned then we can either tweak it until it does work, or throw it out and start again, the worst option is to leave it alone, especially as we know it works elsewhere.
There seems to be a view that free market v socialism is a binary choice. Its a continuum, and both pure socialism and pure free market are equally bad. No country follows either path 100%.
This is also why insulin is expensive (the discoverers and patent holders made their patents freely available long ago). It's regulations which keep insulin, and other drugs, protected from competition and therefore can be expensive... Really, there are endless stories about this.
The gov vs private debate is kind of academic when the reality is a lot of the problem comes down to collusion and cheating from both of them.
Usually competition is the way to insure fair prices.
The collusion then might be to not allow competition. In the case of pharmacy, for example, I would guess there are many regulations that make it very expensive or infeasible to start a competing firm.