Such a categorisation already existsed in the form of the original top-level domain names, but it was never enforced. .gov for government, .mil for military, .com for commercial, .edu for educational facilities, etc.
For good reason, though. Who says what information is "verified", the government, the news media, the publishers? And what governments, media, and publishers get a say?
Perhaps it'd be nice for browsers to show a little indicator to confirm that a website is hosted by a government (although .gov and .mil are only valid for American governments, government could use a given domain as their government basis). There's a big difference between what's right ("climate change is real") and what the government is saying ("who knows, maybe drinking bleach is a good idea?").
Dividing the net goes straight against the idea of the internet. I don't think using a special browser for special domains is very tempting. We'd probably end up with the alt-net version of onion.to to proxy all different kinds of sites to a single application.