I think the way the law was written kind of eliminated that "out" that was there. It did not matter where the server was based, but more the owner of it. For someone like me in the US, I don't have to worry about it.
And no, there were some pretty popular sites shutting down.
Incredible as it may seem, thanks to the Online Safety Act, dozens of harmless, safe, small websites are closing down by 17 March, rather than face threats of fines that could lose their operators their homes.
www.openrightsgroup.org
Free-to-play text MMO browser game, Urban Dead, will close just shy of its 20th anniversary because of the introduction…
www.gamesindustry.biz
news.ycombinator.com
There's been a lot of discussion on it. A few folks that I know that live in the UK are still debating on whether they even want to keep running a site due to the requirements and the possibility of punishment if they don't do perfect. And no, none of them run big sites, but should it matter whether the site is large or small?