Am I wrong, or is the Democratic party just larger than the Republican party and therefore more mainstream?

Generalizing over the whole nation, the higher the voter turnout, the better the Democrats do. This is part of why the two parties like to accuse each other of tolerating voter fraud or illegal voting (Republicans accuse Democrats of this) and of suppressing voter turnout through tricks such as voter caging and inconveniences such as short polling hours (Democrats accuse Republicans of this).

Equivalently, Republican affiliation is more common in sectors of the population that are more likely to vote — e.g. the retired elderly.