How to make it work? Quick thoughts.
Remember that in China, they try very hard to keep people using their own search engines and internal social media systems.
I can see something like a non-erasable cookie or other token containing something that identifies the kid's age, stored on a device that can and is queried by the search engine and social media sites, or even possibly by the CGNAT endpoints that the mobile devices use, to put additional controls on how searches work, and provide age-appropriate per-site IP block filtering on a device. Make the same controls also feed into the network layer in the phone to ensure that the device prevents the use of IP stack based VPNs.
You would possibly have a problem with browsers and other software which have their own VPN built in, but if you make it so that when in this mode, apps are either vetted before installation, only come from 'safe' app stores or maybe are completely blocked to prevent them even being installed. Stop side-loading as well (by forcing it to use certificate based verification, maybe).
Going one stage further, you could put the user's ID information including birth date into all phones sold in these regions (not just kids), and prevent any phones without this information connecting to the national infrastructure. Make strong biometric authentication (linked to a national ID system maybe) mandatory on the phone. This allows even adult phones to be put under some control, maybe just not so much (and may also allow peoples ID and movement to be tracked more easily by making this token remotely readable using mandatory Bluetooth or NFC).
I'm sure that there could be ways around this, but like many things, it will evolve over time.
The real secret would be to allow users enough content to keep them happy without them even really realizing that they're being restricted. If you make it pervasive across all devices, they won't learn through peer contact what they're missing out on. When you're an authoritative regime, and can make policies such as this law, many things are possible, and you can then just define people who try to get around the controls 'dissidents', and 're-educate' them.
Mobile phones (and IT in general) are very dangerous technologies for people's freedom, while all the time pretending they're facilitating it. We're all frogs, slowly being boiled.