This is more of Skeptics-style question, but since I only know of people on Stack Exchange challenging this (so it would not be a "notable claim" for Skeptics)... I'm asking here.
For context as what we're talking about, I'll recall that
In epidemiology, the curve refers to the projected number of new cases over a period of time.
In contrast to a steep rise of coronavirus infections, a more gradual uptick of cases will see the same number of people get infected, but without overburdening the health-care system at any one time.
The idea of flattening the curve is to stagger the number of new cases over a longer period, so that people have better access to care.
Some people challenge that social distancing measures like banning public gatherings etc. are truly effective at reducing the transmission of contagious respiratory diseases like influenza. (I'd link you the Politics SE questions on this, but most such questions got deleted over there for being off-topic and the askers told to ask here... which for some reasons none of them seem to do.)
So what is some convincing evidence from past epidemics that such social distancing measures are effective at "flattening the curve"?