Small gov, direct democracy and free speech is what i associate with liberalism. The US rightwingers/conservatives can be at times ultraconservative for eg. regarding abortion but they are way more liberal when it comes to covid restrictions and guns etc. European rightwingers/conservatives are more conservative so they were mostly in support of the authoritarian covid policies. If the opposite of conservatism is radicalism the opposite of liberalism is authoritarianism.

The US has it's history with libertarianism so that's likely the reason why the rightwingers are pro small gov.

Yes conservatives lefties and woke are the same however european politics is way more leftwing. For eg. most of the society accepted lockdowns, 80% of the people are vaccinated etc.. also the populist parties are seen as very controversial unfortunately there are no normal conservative parties so you gotta choose in between rightwing populism or immigration liberalism. Most europeans are against conspiracy theories so that's also big difference.