Almost all major credit cards have a system to generate virtual card numbers with dollar limits on them. Generate a virtual card number with some limit, give to the kid or configure it on their device. Every time you need to increase the limit you have to login on a site that only the parent has access to.
Huh? The old model is working just as well as ever. Why would you give your kid your credit card? Have them make their internet purchases with a gift card, the non-insane way.