How does the US government, force a Russian website, hosted in Russia, for Russian people, following Russian laws, to shut down?

It shouldn't be able to, clearly. Not any more than Russia should be able to shut down a US website, hosted in the US, for US people, following US laws.

Same thing they do to every country, Pinky. Have a small team invade the country and disappear the people they don't like[1].

Or tap the fiber lines at the border and inject RST packets from off-path, which is something the Great Firewall of China does, and is ironically much more transparent than what they actually are doing.

Or cut the cables between the USA and Russia, or between the USA and any country that doesn't cut their own cables to Russia. The USA did this to Iran with the banking system and it worked: the USA cuts money transfers with any country that doesn't cut money transfers with Iran. I don't think it would necessarily go their way if they did it right now with the internet.

[1] https://en.wikipedia.org/wiki/Abu_Omar_case

Dove a bit into this topic superficially out of curiosity. Maybe not shut it down but greatly limit reach:

- Domain Name Seizures via ICANN and registrars

- Political/legal pressure on CDNs, SSL certificate providers, bandwidth providers.

- Propaganda and legal labeling ("malicious actor", "foreign agent", "terrorist")

- There are technical workarounds to keep the page up within Russia's sovereign internet (Runet).

It doesn't.

They shouldn't.