I think one approach would be to toughen laws against "revenge porn."
When it's a literal crime to share explicit material without permission of the people in the material, site operators would hopefully not risk jail time by just taking the word of an anonymous user that they have permission to upload the video they're uploading.
I think this would kill the business model for free porn sites, leaving it to the original producers - the people who actually have all the required releases on file - to be the only ones willing to post the content online, but to actually make a business case for this, they'd need to get their revenue from direct fees to users... i.e. put their content behind a paywall.
This wouldn't make the content completely unavailable to kids, but I'd say that the number of kids with their own credit card whose statements don't go to their parents is pretty low.