As your question suggests, conservatism in the United States was not always associated with a single party. Neither was liberalism, for that matter. The shift toward associating one single party with the ideology started in a large sense during and after the election of 1964, when Barry Goldwater won the Republican nomination over liberal Republican Nelson Rockefeller. Goldwater's platform was centered on taking a harder line in the Cold War and deconstructing the New Deal's liberal welfare state.
The divide between liberal and conservative America was only increased during the election of 1968. Richard Nixon won the election by presenting himself as the champion of the "silent majority." These were middle-class Americans who weren't anti-war protestors or counterculture activists. In fact, many conservative and even moderate Americans were alarmed over the trend of the Supreme Court expanding rights for minorities and criminals. They felt that the cultural shift in the country had gone too far leading to an erosion of traditional values. To halt this, Nixon promised to be tough on crime and be a champion of traditional American morality. Nixon's "Southern strategy" hinged on this and attracted historically Democratic voters in the South as well as racists upset over the Democrats' support of civil rights legislation.
This move clearly established the Republican Party as the party of conservative America. In the elections since 1968, Republicans have leaned into this identity by presenting themselves as the champions of small government and traditional American values.