Conservatives lose culture war battles over controversial speech and censorship because they tend to hide their views behind lofty language defending the principle of free speech, while progressives make their views clear and critique the substance of what they dislike.

An example of this would be Joe Rogan right now. Spotify employees want to boot him for being transphobic, and they cite specific comments he made. Rogan's defenders will respond by saying "it's his right to free speech!" but avoid talking about the actual trans debate at hand.

It's obvious to any neutral person with no dog in the fight that Rogan's defenders don't *really* care about free speech--they just agree with Rogan about trans people.

This lack of forthcomingness makes Rogan's defenders look dishonest, and makes the progressives look better.

The side of a culture war that is appealing to higher ideals is the side that's losing. This also plays out in the rare instances when conservatives are on the popular side, e.g. late-term abortions. Liberals appeal to bodily autonomy, while conservatives directly call it murder.