What separates the left, especially the European version of left, from the American concept of left/liberal?
Left=pro-worker=socialist, though that word has been poisoned by a century of Red Scares here.
Right=pro-aristocrat=capitalist. (When you have a hereditary class of state-backed "owners" who control the economic resources, that's an aristocracy.)
Left/right has nothing to do with social issues like abortion rights, gay marriage, etc., nor does it have to do with the size of government. It's about who the economic system should benefit.
The Democratic party is thoroughly capitalist, with only a relatively minor disagreement with the Republican party as to what degree of exploitation of the proles by their feudal lords is best.