Hubble resolved stars in the nearest galaxies and measured their distances using Cepheid varibles in the 1920’s and together with Slypher and Humason established that the universe was expanding in the 1930’s. By 1967 the Big Bang was firmly established as the prevailing theory since the measurement of the cosmic microwave background by Penzias and Wilson in 1964
The turning point is often dated to the “Great Debate” of April 1920, when Shapley and Curtis publicly debated whether the “spiral nebulae” — i.e. what we call galaxies today — were part of the Milky Way, or further away.
In fact, the debate only really gained its historical significance a few years later, around 1924-5, when Hubble demonstrated the spiral nebulae had to be much too distant to be part of the Milky Way.
The significance of the Great Debate is that it was the last time that anybody could argue against the existence of external galaxies without being obviously wrong.