So, this is a very complex topic I don’t have the time to give the treatment it deserves, but to try to give a very summarized historical viewpoint on it -

Liberalism was a set of ideas that cohered around the 18th century as a reaction to monarchism that emphasized universal civil rights and free markets (there were a ton of weird things going on with noble privileges and state monopolies issued by royal administrations and mercantile economics this was a response to)

Socialism was a set of ideas that cohered around the 19th century as a reaction to liberalism (and the whole industrial revolution) that said universal civil rights didn’t go far enough and we needed to establish universal economic rights. Some socialists think the only way to achieve these things is by overthrowing or limiting the power of governments and ripping up contracts between private parties, which liberals tend not to like.

Progressivism was (sort of, I’m being very reductive here) an attempted synthesis of these traditions that cohered around the early 20th century, and (essentially) argued “ok, free markets but restricted by regulations (e.g. you can’t sell snake oil, you can’t condition the sale of property on the purchaser being a specific race), and open elections for whoever the voters want but with restrictions on the kinda of laws that can be passed” (e.g. no poll taxes).

Like I said, I’m simplifying a lot here and I’d encourage reading Wikipedia pages and other sources on all of these things (like, I’m eliding a whole very dark history progressives have where their attempts to perfect society had them advocating for eugenics and segregation early on because there was academic support for those ideas at the time, and there’s a lot more to be said on how a lot of the first anti-racist voices were socialist ones and why it took progressives and liberals time to get on the right side of that issue, and how fights for colonial independence tended to be led by socialists and against liberals), but the fact that liberals progressives and socialists are all ostensibly “on the left” is a big cause of the infighting we see.