As Robin says, the answer is the 1920s.
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.