They are not just out of the blue intuitions though, they are based on millennia of observations.
Empiricism is not just found in scientific labs.
I am no expert in the history of social norms, but I don't think it's clear at all the protecting children from witnessing violence and sex is a human universal. I know of several cultures (e.g. Roman Empire) where it was normal to let children witness public executions or violent fights. Again -- I am not arguing that this is good, I am arguing that one should not make laws based on vague intuitions.
Most people didn't make laws and force social norms either though. The "educated" classes that did had theater and other kinds of public shows (music, etc) since at least the time of Ancient Greece, and those did have certain codes about what is to be said and what shouldn't be said.
They also had special "adult" shows.
None of these cultural norms are that new either. Even back in ancient Greece and Rome there were rules about what constitutes public or on stage obscenity.
You can argue that the internet solves this, but it's not really the case: for example consider the current YouTube "demonetization crisis" or the arbitrary rules imposed in walled gardens such as app stores. The bigger danger -- and there are many examples of this -- is that "think of the children" can be used by politicians to advance more nefarious censorship agendas. For example, trying to rate content that presents non-conventional ideas.
Please read again. The second paragraph is precisely where I say that I have no problem believing that there might be ill effects. Our intuitions, however, are no replacement for scientific rigour. The history of science if full of cases where common sense turned out to be wrong.
More importantly: I am not proposing that a child should be subjected to "nonstop violent porn". That sound like a terrible idea and it is a straw man. Like most things in life, this is not a binary choice. It's not like we have to choose between puritanism and nonstop violence and porn. What I am saying is this: is catching a glimpse of violence or sex so damaging that our entire culture should be subjected to severe restrictions? Or is it something best left for parents to decide? Maybe some parents are more worried than others, and maybe it doesn't make sense to burden everyone with a certain standard -- unless it is supported by rigorous scientific research.