All health insurance plans in the US are private-market.* You may be confusing ACA-compliant plans (private insurance that's allowed to be listed on healthcare.gov and is eligible for subsidies) with non-ACA-compliant plans (not allowed to be listed on healthcare.gov, not eligible for subsidies).

Be careful about non-ACA-compliant plans. They're cheaper because they engage in sketchy behaviors that leave you without coverage when you need it.

*Well, except Medicaid, Medicare, etc.