Perhaps you could explain this more? It seems to me that many first-world countries regulate much more than the United States (ie single payer or similar) and have lower costs.
Not all regulation is born equal. When the single payer is also the regulator (which is the case for nationalized healthcare), regulation is introduced to limit the costs. When the regulator is being reelected thanks to the money given to it by the players, there's much more variance in the outcome of the regulations.
There's a lot of bad regulations in the US, on top of my head: public hospitals are not allowed to negotiate the price of medical equipment, so pay 5-10x the price paid in France, for instance. Gross margin of health insurances is capped as a % of the underlying medical cost, so the higher the costs, the higher the profits.
US healthcare system is simply crazy. I (and my family) have very very good coverage thanks to my current job, but I'm out of the country as soon as the situation change (not that I'd have much merit doing so: my visa doesn't allow me to change job, so if I'm fired or my employer collapse, we will have to leave anyway).
Inflation of healthcare costs is directly caused by government regulation.