I think as much as people would like parents to be omnipotent, including parents themselves, I think they could be helped more.
It's not possible. We cannot police the internet. It's too big, there's too many jurisdictions, different legal systems, hidden areas. It's physically not possible.
Parents have to monitor the usage of one or two children, usually. There are systems in place to help with that - parental controls, activity reports, etc. Neither of those require the parent to be omnipresent or able to watch their child online all the time.
Monitoring the whole internet would be incredibly difficult. To be honest, when someone finds a good way to do it, it'll be used for detecting crimes, removing manuals on how to make bombs, removing child abuse imagery, identifying abuse circles, identifying potential terrorists etc, LONG before it's used to censor the internet for children.