Some things to consider:
1. This is true sometimes, but still privatization and insurance's fault. Hospitals have a motive to charge insurance more money. Public healthcare doesn't have this problem.
2. Administration and billing is ludicrously expensive because of insurance. Thousands of insurers with their own processes breeds the most inefficient and broken system you could possibly imagine. The solution is removing the market, i.e. having a single payer. Boom, administration costs disappear.
3. Preventative care is rarely paid for by insurance. In addition, many patients simply don't do it. It's easy for a smoker to get treatment for COPD, it hard for a smoker to stop smoking. That's just how it is.
No matter how you slice the healthcare problem in America, it undebatable that private insurance must go. The inefficiency it creates is absurd and it's why our healthcare costs are through the roof.
Not only do we pay much, much, MUCH more per citizen for care, but our care is also lower quality as compared to the rest of the west.