Well there is the US libertarian party and Libertarianism, they are similar but would give you very different answers to your question.
Libertarianism is a political ideology, more of a theory of governance "guidelines" if you will. Where the libertarian party is a platform that looks towards Libertarianism as well as free market capitalism amongst other ideologies to come up with a party platform.
So with your question
child labor laws? anti-discrimination laws? mandatory work weeks? workplace safety protections? At what point does the capitalist magic happen, where all the nasty exploitation in the contemporary system disappears and the free market starts to take over?
"Libertarianism" Would say that so long as all those things listed are done voluntarily what authority do you, or any government have over the individual to prevent them from entering into any of those contracts? If it is not voluntary then your rights are being violated and those actions should be illegal.
The libertarian party would have laws about child labor, but probably much more relaxed than they are today. Some states say you have to be 16 to work, but some kids may want to work at 15, why should they be denied access to employment simply because of their age? Or children helping with family businesses or family farms? How does the government acquire the power to deny your child from voluntarily working on their family farm? And again, it would have to be voluntary, a parent has no more right to force their child to work as does a corporation have a right to force you and I to work. Children would have rights to voluntary association.
I believe that both big L and little L libertarians believe that anti-discrimination laws are in fact discrimination laws in and of themselves. It's just that they discriminate against the "bad guy" so most people support them. Freedom of voluntary association trumps all that. You have no right to my labor, I have no right to your money. For you to get my labor for your money we must agree, and agree without force or coercion. That being said there can be no "pro" discrimination laws like Jim Crow. If a business want's to only cater to women, they have that right. If a radio want's to be "black owned and operated" they have that right. If the KKK says "no blacks allowed" they have that right. There is no
Just keep in mind that while the market has proven good at certain things - it has an absolutely miserable track record when it comes to the treatment of workers.
This is true, and it's something we need to work on as a species. We treat each other like shit most of the time. I don't think any political philosophy can solve inherent human problems. But political philosophies can create the 'atmosphere' for it. Look at the industrial revolution. There were so many laws being broken and so many rights being trampled on by corporations. The problem wasn't that there weren't enough laws, there were plenty of property laws and personal laws against what these corporations were doing. The problem was that there was no Justice. The federal government in an attempt to better compete with Europe basically told large business "do whatever you want, and you will get away with it" so the big businesses did. If there was actual justice back then most of those things wouldn't have happened, or at very least would have happened a lot less.
Even if your entire country only had 3 laws; You can't harm any other person, you can't harm any property, and you can't steal. And only those 3 laws were enforced, think of all the things that wouldn't have happened (or even continue happening (BP)). What happened in the past was not a result of too much liberty, it was a result of federally protected businesses being allowed to violate the rights of individuals. (Think tort reform to the most epic scale)
At what point does the capitalist magic happen
Very idealistic, and I wouldn't call it "magic". And I don't think you can quantify it that way.
If you were not only "allowed" to be free, but your freedoms were actually protected by your government how long do you think it would take for your life to be better?