Very few US prisons are private, only 8% of prisoners in the US are in those facilities. Some states have recently ended the practice, so I would expect to see that number trend down in the future.

You don't need private prisons per se in order for the perverse incentive to exist – even thrive and persist. The beneficiaries of prison labor are private interests, whereas the cost of incarceration is paid by the public. It could cost 100k per inmate a year in public money and produce 30k of goods. It's still (virtually) free labor as far as the beneficiaries and their lobbyists are concerned.

I’m not in favor of compelled labor, or private prisons. In general I think the ideal would be for the taking of one’s freedom by the state to be rare. I’m merely pointing out that private prisons are a tiny fraction of the whole pie, and almost certainly not “The Problem”.