> One famous example is monopolists that are created in a free market and then use their power to make the market unfree. Only government regulations destroy these monopolies with anti-trust laws.

The thing about "free market" is that it's like "reasonable person" in that it's an ideal rather than something you can actually find in the street. You want to get closer to it even if you can never actually find a flawless example.

A monopoly is basically the opposite of a free market. But where does a monopoly come from? For the most part they're a result of government regulations. The monopoly sucks and everyone hates it so you want to go into competition with them, what stops you? For the monopoly to persist there has to be some law that prohibits anyone from doing that or makes it infeasibly expensive.

The problem here is that unless you're going to go ideologically pure full anarchist, you can get to that point if the government does as little as enforcing contracts or property rights. Because then you could have contracts for selling your children into slavery or a single corporation that the government says owns all property.

So once the government gets involved with something, it has to be sure nobody is leveraging that into a private monopoly. But this is why government involvement should be minimized -- that risk increases the more they do, and there is abundant evidence that governments repeatedly fail to mitigate it effectively and prevent private monopolies from forming.

> Then there is pricing in so called free markets that are not in fact free. Such as pricing of infrastructure and services.

You're referring to natural monopolies, which is once again obviously not an example of a competitive market. The key in these cases is to narrow the natural monopoly as much as possible and prevent anything more from being tied to it than is absolutely necessary. For example, one of the classic mistakes we've made is allowing the natural monopoly on the physical last mile for data service to be leveraged into control over interconnection and over the top services like video and telephone.

> To name one perfect example of where government regulations lower the prices of services is in the healthcare industry. In most western countries (not the us tho) healthcare is relatively cheap and also accessible. This is because it is regulated. In the US it is barely regulated and very expensive.

The US healthcare industry is one of the most highly regulated industries in the world. Competition is low because FDA approval is hard to get and the AMA uses regulatory capture to limit the supply of doctors, government incentives promote employer-provided health insurance that disguises pricing from patients, "certificate of need" laws constrain competition between healthcare providers etc. etc.

The typical claim is that single payer systems have lower costs, but nearly everything has lower costs -- the US has some of the most expensive healthcare in the world -- because it has some of the most onerous healthcare regulations. Meanwhile countries like Singapore have lower costs and good outcomes with a predominantly private healthcare system, because it is less inefficiently regulated.

> Without regulation the market will also destroy the environment.

This is clearly not an example of regulation lowering consumer prices. The claim was never that regulation can never be necessary. You don't want the widget factory dumping industrial waste in the river, but prohibiting that isn't generally a means to lower widget prices.

It can even do both things at the same time. You can have a regulation that succeeds in reducing pollution while also being dramatically less efficient than some other means of achieving the same goal. This is why government regulation is so precarious: If you get it wrong nobody else can fix it because anybody who tries to do it another way is incarcerated.