>Aren’t a lot of jails private and for profit?

No, it's a small-ish minority of them. Most are government owned and run.

That said, there's a huge incentive to piss away money holding people so you can justify your budget and use poor conditions to justify increases in budget. And on top of that the contractors that supply government jails are pretty evil too.

So it's really a distinction without a difference at the end of the day, it's all a pretty rotten system.