There are a couple of methods for filtering the Internet for your child. One is to essentially block all sites and only whitelist the ones you want your child to visit. Young children will probably only have a few sites they like anyway. The second method is to use a more traditional filter to block sites you deem inappropriate. Filters like these are not 100% effective, but they should prevent a lot of accidental visits to inappropriate sites.
An example of a Firefox extension that can be configured to use either method is ProCon Latte.
Like desktop restrictions, an Internet filter will not stop older children from purposely visiting sites you believe are inappropriate, but it will reduce accidental visits.
Aside from filtering content, you can also filter ads with browser extensions. This will prevent children from following ads that may lead them to sites asking for personal information. Adblock Plus is available for Firefox and Adblock for Google Chrome.
Kids ultimately will use their computers for fun and learning, so here is a brief list of games and learning activities for kids. It is not a complete list, but should give you an idea of the wide range of games and activities available for children.
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As you can see, with just a little work on your part, Linux can be educational, safe, and fun for your children.