Say you logged into your Amazon account and set the rating level for content available in Fire and then Amazon's servers would block your kid from accessing freepr0n4all.com. You could add exceptions, specific sites to block, etc. and there would be a known, ever-evolving database of "unsafe" sites by rating.
The ability to filter content in a mobile browser isn't something that exists as of yet, I don't believe.
If I could do this, I'd buy one for my 9yr old today.
Of course, this could then be combined with other tools to "lock down" the fire, e.g. disabling the ability to exit "cloud mode" and thus bypass the filters, password protection for app/movie purchases, like iOS parental controls do, and so on.