Michael Cannon at Cato explains this fallacy:

Advocates of a new government health care program also claim that government contains overall costs better than private insurance. Jacob Hacker writes, “public insurance has a better track record than private insurance when it comes to reining in costs while preserving access. By way of illustration, between 1997 and 2006, health spending per enrollee (for comparable benefits) grew…

[But] Hacker’s comparison commits the fallacy of conflating spending and costs. Even if government contains health care spending better than private insurance (which is not at all clear), it could still impose greater overall costs on enrollees and society than private insurance. For example, if a government program refused to pay for lifesaving medical procedures, it would incur considerable nonmonetary costs (i.e., needless suffering and death). Yet it would look better in Hacker’s comparison than a private health plan that saved lives by spending money on those services. Medicare’s inflexibility also imposes costs on enrollees. Medicare took 30 years longer than private insurance to incorporate prescription drug coverage into its basic benefits package. The taxes that finance Medicare impose costs on society in the range of 30 percent of Medicare spending. In contrast, there is no deadweight loss associated with the voluntary purchase of private health insurance.

Read more: Private Insurance Is More Efficient than Medicare–By Far.