Liberal across the Atlantic means something closer to its classic definition.
One of the first challenges I have when discussing politics with Americans (and increasingly other Canadians) is precisely trying to define how I, as maybe a more "traditional" socialist radical, do not fit on the spectrum they've defined there. I share a disgust with the conservative right, am secular, and broadly call myself a feminist or anti-racist but when it moves to bread and butter issues, gun control, or some identity politics stuff... there's basically no correspondence...
(Here in Canada, it's muddled in different ways, where "Liberal" is a political party, defined mainly by "whatever will win an election" than any constant ideological focus. Right now it's vaguely "left" (in so far as it takes "left" positions on cultural issues, combined with some kind of neo-Keynesian budgetary spending model, but combined with low corporate taxes) but in my mind it's really just a kind of croneyism... but that's a rant. We have a separate, sometimes-successful, social democratic party which sorta muddies any kind of comparisons to the US, but comparisons to the UK also don't work because our NDP isn't really like their Labour party, either... though it might want to be)
Incidentally, if you talk about liberalism people are more likely to understand that you mean the philosophy rather that some basket of policies and jargon that are popular with the liberal tribe at the current moment.
> broadly call myself a feminist or anti-racist
Even these labels are fraught, at least in the US. They are pretty much doublespeak insofar as the folks who identify as 'anti-racist' in the US tend to endorse policies that are explicitly racially discriminatory (they would say 'race conscious', but they mean exactly the same thing), and the folks who identify as 'feminist' in the US have no discernible interest in gender equality.
This was not some grand conspiracy, the Socialists just realized that workers in the West were generally happy and well off (iirc Marcuse said "Capitalism delivers the goods") and not at all friendly to Socialists' plans of revolution.
As a result there was a push to shift the propagandizing ("educating") to the next best oppressed groups they could find, with these being the familiar categories of LGBTQ, racial minorities (only certain ones), women etc.).
Just like the socialist revolutionaries of old, the majority of the "New Left" are not parts of those groups. They are merely using them as a "revolutionary mass" whose misery and grievances are single-use tools for the revolutionaries to gain power. Lenin was fuming at Bolshevik party members who actually tried to negotiate better working conditions or pay pre-1917 because happy workers don't have "revolutionary potential" (a euphemism for not being desperate enough to want to put him in power).
> If, as I keep hearing from right-wing commentators, "the left" is somehow dominant, it's only because "the left" has wholesale dropped its historical anti-capitalist mission, and has retreated almost exclusively into culture wars; and generally taken positions which enrage the right on that basis, but doesn't actually challenge any real power.
I explored some of the reasons for that above, but everybody with some sense could see that capitalist societies were happier, almost never genocided their own population, were vastly more prosperous and even produced enough surplus to have better social welfare than Socialist countries. On the other hand Socialism always ended badly, sometimes catastrophically. People are not blind, and associating your political movement with 60 million corpses is not a smart move.
The reason they don't challenge "real power" is because the objective is to gain the power vested in those institutions, not to challenge them.
And the milieux that circulates this is really primarily liberal, centrist in practice, overwhelmingly supporting and electing quite mainstream fully "capitalist" political parties... but with the aesthetic or appeal to a past radical, anti-capitalist, socialist "scene"... without an actual meaningful program or alternative for the working class. A working class which has as a result almost completely divorced itself from this "left."
Which is all the end product of the demolition of the organized socialist left & trade union movement in the west through the 70s and 80s and early 90s and the retreat of associated intellectuals into academia and cultural politics.
If, as I keep hearing from right-wing commentators, "the left" is somehow dominant, it's only because "the left" has wholesale dropped its historical anti-capitalist mission, and has retreated almost exclusively into culture wars; and generally taken positions which enrage the right on that basis, but doesn't actually challenge any real power. A very general "anti-oppression" politic which is primarily concerned with individual concerns, not collective ones. This "left" that I keep hearing about from the Shapiro etc types foaming at the mouth, it's a caricature not only made by its opponents... but by itself.
Just Liberalism, and a very 21st century American product.