The banking system was creating money in various forms outside of QE. That was just a well known mechanism.
The main way it's created is through loans, many of which end up in housing. And rampant speculation on apparently ever-rising house prices is where the financial crisis started. Ripples expanding ever outwards ...
The last round of quantitative easing by the Federal Reserve, QE3, ended in 2014. (QE1 began in 2008.) If these price effects for health care and housing began in 2008 and ended in 2014, it would make sense to blame QE, but they didn't, so it doesn't.
Price effects that are associated with government vs. private expenditures are not the same as inflation. That's just governments being bad (perhaps intentionally bad?) at spending taxpayer money efficiently. However, when it comes to health care in particular, that just isn't the case either--Medicare and Medicaid pay much lower prices for medical procedures than private insurance companies do.