These terms by now can mean almost anything you want them to mean.
I tend to view liberalism as a right-wing ideology, basically running together with capitalism. “Liberals” and “conservatives” in the US argue about parameters of state policy within the context of a state capitalism. This isn’t just a theoretical or philosophical observation, it’s also a practical and empirical fact in the world. In Australia, the Liberal party is the conservative party. In Western Europe, to be “liberal” is to be on the centre-right, i.e. conservative.
So the right, as far as it is concerned in economics, and so far as it expresses itself ideologically (the reality is often different), has been split between those who advocate less (conservatives, capitalists, “free-marketeers”) or more (liberals, Keynesians) State activity in the economy
On the left, I would place anarchism/marxism/socialism and some forms of social democracy, though most social democratic parties have long betrayed their socialist roots and become identified with various forms of class compromise; their various welfare state structures have long been under attack and have been rolled back, often by the social democratic parties themselves. But the decline of social democracy is another long question and less relevant to the US.
Historically, I think one can basically understand this, at a ridiculously broad level of caricature, by noting that “history moves left” in the following sense.
1800: Democracy/capitalism/liberalism/left vs Monarchy/feudalism/conservatism/right
1900: Socialism/Social-democracy/liberalism/left vs capitalism/political-democracy/conservatism/right
1950: Socialism/Keynes/”modern liberalism”/left vs. capitalism/Hayek/”traditional liberalism”/conservatism/right
2000: Socialism/”neo-Keynesianism”/left vs capitalism/liberalism/conservatism/right
And, extrapolating, based on this I fancifully postulate an optimistic future history.
2050: Anarchism/Socialism/left vs social-democracy/capitalism/right
2100: Anarchism/participatory-economics/left vs market-socialism/social-democracy/right
2150: anarchism/left vs participatory-economics/right
Of course, history is full of setbacks, regressions, and catastrophes; and there is nothing deterministic about the future of humans, much depends upon will, intention, and political and social dynamics at every well.
Still, I would like to hope that if I live to 2150, then I will have to think about whether or not to join the conservative party.
Let us realize the arc of the moral universe is long but it bends toward justice.
— Martin Luther King Jr.