Not entirely true, some states like Montana have half of their prison population in private prisons. These prisons are paid by the government to keep people in prison, there is no economic incentive to eliminate recidivism.
"incarceration not only upholds the justice system, but also subsidizes construction companies, companies that operate prison food services and medical facilities,[5] surveillance and corrections technology vendors, corporations that contract cheap prison labor, correctional officers unions,[6] private probation companies,[5] criminal lawyers, and the lobby groups that represent them. The term also refers more generally to interest groups who, in their interactions with the prison system, prioritize financial gain over rehabilitating criminals."
While the government may not be directly profiting in public prisons, this interest bloc is capable of lobbying and does do so.
en.wikipedia.org