Watch with your child. Simply ask your child what they're watching and join them. In general, children are tuning into certain channels or following specific YouTube personalities because they're entertained by them.
Watch by yourself. If children don't want to share, get the name of the channel they're watching and watch it later. Watch a few videos by the same creator to get a feel for the content.
Be 'sleuthy.' If you're concerned about the content your child is watching on YouTube and you've tried talking to them, there are ways of tracking their viewing habits. If they have a YouTube account (which only requires a Gmail address), their YouTube page will display their recently watched videos, recommended videos based on their watch history, and suggestions for channels similar to the ones they've watched. Even if your child deletes their "watch history," the recommendations all will be related to material they've watched.
Subscribe. Encourage your child to subscribe to their favourite channels rather than hunting around on YouTube for the latest ones from a specific creator. Subscribers are notified when a new video is uploaded, plus all their channels are displayed in My Subscriptions that you can easily access.
Look at the suggestions. The suggested videos listed on the right-hand side of the page are related in some way to the main video. Evaluate them to see if they seem age-appropriate, and that will provide an indication of the appropriateness of the main video.
Read the comments. YouTube comments are notorious for being negative, but it's worth reading them to get a sense of the tone of the discussion. If your child posts video clips on YouTube it is a really good idea to ensure that they deactivate comments for this very reason.
Turn on safety mode. Be aware that YouTube is officially only for teens 13 and up, and what the site considers age-appropriate may not match your values. But YouTube offers a filter called Safety Mode that limits the inappropriate content. The tutorial in two weeks will show you how to implement this. It is well worth the few minutes it takes to set this up.