One major battle ground has been sci-hub. This is best described as a collective whose stated goal is to make science free for everyone rather than having it stuck behind a journal paywall. The entire journal industry is one giant scam, but I digress. U.S. laws have repeatedly been passed and tested, and Google has been repeatedly ordered to remove domains such as sci-hub.tw and sci-hub.cc, among many more.
This is far more expansive than scientific papers, of course, and extends to countless copyright adjacent websites. Sites which have been repeatedly targeted include YTS, FMovies, and AniWave (and their associated domains).
I should also expand my previous comment. Google doesn't just censor results which the U.S. government has requested. It very frequently censors results of its own volition. Whether for ideological or fiscal reasons. One litmus test I often use is to search for the Proud Boys. Their website(s) have long been censored for a variety of reasons, including "disinformation" and public safety. I can find their website on Kagi. I cannot find it on Google.
Whether by the government, or for ideological or fiscal reasons, when I search for information I expect to be given that information. Google is an unreliable source. They all are.