Yes, American conservatives are basically classical liberals whereas American 'liberals' are actually social liberals:
Social liberalism - Wikipedia
IMHO, what American political pundits refer to as
"liberalism" (the Democrats, the 'left') and
"conservatism" (the Republicans, the 'right') in popular discourse, are both 'sects' of enlightenment era liberalism - their shared secular creed - or rather contemporary mutations thereof.
By analogy, they can be compared to denominations within Christianity - like Catholicism and Protestantism.
America was inherently conceived of as being a liberal republic (or, at least, functioned as such prior to Trumpist populism infecting the GOP), so it should be unsurprising that its main political parties are both competing schools of liberalism.
Both Republicans and Democrats, therefore, agree about the need to keep America as a republican system without a monarchy or aristocracy; they have regard for the rule of law and due process under a constitutional system with separation of powers between the three branches (executive, legislative and judicial); both sects believe in the bill of rights and personal liberties such as freedom of expression or religion and a good many other things which separate both of them from autocratic ideologies.
'Conservatives' are the more classically liberal - they are 'conserving' root liberal ideas without much in the way of evolution or cross-pollination with other sociopolitical traditions, such as Marxian socialisms - and so they strictly apply a more laissez-faire free-marketeer ethos in governance, believing that intervention by public authority should be kept to a bare minimum rather than potentially compromising the freedom of citizens with greater bureacracy.
'Liberals' are in point of fact "social liberals" - they combine classical liberal ideals of individual autonomy, freedom and limited government under the rule of law with governmental intervention in both the market and society for the common good, for instance through social insurance programmes, workers rights legislation, healthcare and other areas. In this, American liberals have adapted liberal ideas to respond to valid criticism from Marxists and Socialists in respect of income inequality and social exploitation in liberal societies, imbibing some of their ideas into a refined capitalism without actually challenging the underlying structures of the system (in the way that Socialists do).
American conservatives are liberal purists (impolitely, I'd call them reactionaries), American 'liberals' are liberal progressives.
In continental Europe, "
liberal" is understood and applied differently to how it is in America (I would say more accurately): its a term reserved for 'classical' free-market liberalism (akin to American 'conservatism'), while "
conservative" is largely a word used for Christian Democratic parties which tend to be economically and fiscally centre-left (sometimes just centrist) but conservative as regards social mores/society and "
socialist" is never misapplied to any kind of liberal (as in American parlance, where right-wing people often wrongly refer to social liberals as "socialists") but used only for actual socialists or sometimes very left-wing social democrats.
Britain is influenced by both the continental European and American terminologies, leading to a confusing situation where you have a party of Liberal Democrats who can be both neoliberal free-marketers or centre-leftists, a 'Conservative' Prime Minister Margaret Thatcher who was actually a reactionary classically liberal market fundamentalist rather than a paternalistic Christian democratic conservative (along the lines of Disraeli or One-Nation Conservatism) and a Labour Party that from the Blair years on has ceased being strictly socialist and now has a large centrist, social liberal wing.
In point of fact, Socialists are not liberals of any hue or variety (so the terms should, ideally, not be conflated as they so loosely are in the United States), they hail from a completely separate political tradition/family which only intersects with liberalism to the extent that social liberals have adopted more government intervention and redistribution programmes within a liberal capitalist system, as a response to Marxian critiques of the inequities of liberal societies.
