My kid figured out that if they switch off WiFi and use their data plan instead, all the filters I set up just stop working. Is there a way to block certain sites that follows them regardless of how they’re connected, or am I basically stuck having to go through the carrier for that?
Yeah this is like the classic kid hack lol, they all figure it out eventually. 
The trick is to use a monitoring app that installs directly on the device (like Bark, Circle, or Google Family Link) rather than just filtering at the router level — those work on both WiFi and mobile data because the filter travels with the phone, not the network.
Google Family Link is honestly the easiest free starting point if it’s an Android kid — it handles content filters regardless of connection type, and your carrier does usually have add-ons too if you want a backup layer.