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This was not an action simply fought by movie studios lawyers across the sea. MPAA lobbied the government to use soft power and the government did.

Wikileaks diplomatic cables showed US government put a ton of pressure on the Swedish government over the Pirate Bay. If Sweden failed to act they risked political consequences including being placed on a list of countries that fail to protect us intellectual property and potential economic consequences

Further the 2006 raid on the Pirate Bay was in some part due to US government pressure and while no US officials were physically present they were coordinating with Swedish authorities

American government serves business interests. It's more or less entirely beholden to private interests, even at the expense of the public.

This is why many Americans are so sensitive to state authority: there's no authority in public interest--not even public health, so there's nowhere else left to go but fascism.

This is just copy and paste from Google Ai but it sums it up:

In the early 2000s, the U.S. government applied pressure on Sweden to address the issue of illegal file-sharing, particularly the file-sharing website The Pirate Bay. This pressure involved direct communication, as well as the threat of trade sanctions and potential World Trade Organization action if Sweden didn't take action against The Pirate Bay. The U.S. was concerned about Sweden's role in harboring a major site facilitating the unauthorized distribution of copyrighted material.