And that is exactly the point.
The American prison system is working exactly as intended. It was originally designed to perpetuate the institution of de facto slavery, after de novo slavery had been explicitly outlawed. Which was also the reason that prisons have always had a disproportionately high rate of incarceration for minority populations.
At a time when prisons were undergoing substantial reforms to reduce this effect, to enact education and job-skills training programs, eliminate the use of prison labour, and reduce overall recidivism, the Private Prison Industry was founded to bypass those reforms, and take over the role of slave master. It's not by coincidence that this happened concurrently with the biggest escalation of the War on Drugs in its entire history, and the largest mass incarceration of non-violent offenders since the Reconstruction days. It was shortly to benefit from the widespread adoption of "Three Strikes" laws, which though its lobbying efforts it helped to support.
This period also effectively cemented the school-to-prison pipeline for black and other minority communities.
Dropping crime rates, liberalization of drug laws and elimination of Three Strikes statutes, widespread abuses within the system, and the failure of reforms in many states to end prison labour in publicly-run institutions, kept private prisons from becoming a much bigger industry and taking the lion's share of inmates. As it stands, those that remain typically have contracts that mandate a minimum inmate population, and their lobbyists still work hard to push privatization. A lobbying effort which has seen some success, as the Trump administration has rolled back the Obama administration's push to eliminate federal use of for-profit prisons.
If I understand your narrative correctly, the description you give in the last paragraph undermines that given in the first few.
But it shouldn't be too hard to check.
How many private prisons are in the US and what percentage of inmates do they hold?
How much money does prison labor produce (I think it was "slavery" wasn't it?) compared to how much prisons cost to run?