You could replace Chrome with
Chrome Beta, which is the version of Chrome in beta-testing format that has numerous features not yet available in the standard Chrome version. A while back, a "supervised user" feature was announced for Chrome that could allow someone to control what other users could access across Chrome, but I'm not sure if that feature is on the Android Chrome Beta app yet. Here's a link to the AC article that talked about it:
'Supervised Users' feature coming to Chrome Beta channel | Android Central
If Chrome Beta does work for you, you'd probably want to disable the main Chrome browser so that your son would only have access to Chrome Beta. To disable, go to
Settings>Apps>All, select Chrome, and tap Disable.