I'm worried about the related videos that show up next to what my kid watches on YouTube. Is there any way to filter out what shows up in that section?
You can't remove YouTube's related videos, which display in the Up Next section. But you can adjust some settings to reduce the risk of your kid seeing inappropriate related videos. Google uses different kinds of data to try to tailor the videos in the Up Next section to your taste, including the video that you're currently watching and even your search history across other Google products, such as search. But this method isn't perfect and allows some videos you don't want to see slip through. Here are some tips to reduce inappropriate related videos and tailor them to your taste:
Turn off Autoplay. The related videos will still display, but they won't play instantly.
Tell YouTube what you don't want to see. Click the three dots next to the Up Next video title and select Don't recommend channel. That will take the channel out of your feed. (This feature is rolling out in July 2019, so it may take a bit for it to hit your devices.)
Manage History. Go to History in the Library section. Toggle off your Watch History and your Search History. This erases all of the previous data Google was tracking to identify related videos and lets you start fresh. You can also delete individual videos from your history when you don't want Google to use them for related videos.
Use Restricted Mode. Go to your YouTube account settings (the little gear symbol). At the bottom of the page is the option to enable Restricted Mode. This will cut down on inappropriate videos but won't eliminate them completely.
Give Google Feedback. In the Up Next section, click on the three dots that display next to the video. Select Not Interested and then Tell Us Why.
Subscribe to channels you like. When you subscribe, you'll see these channels in your feed, and it also helps YouTube's algorithm predict what you'll like.