Cuomo says that doubling rate of new hospitalizations has stretched out from 2 days on Sunday to 4.7 (implied) on Tuesday, which would suggest that social distancing is already slowing the growth. That would put us well ahead of Italy in getting this under control.
All you can do by slowing spread is putting us back to exactly where we were when there were fewer infected - ending isolation restarts spread. There is no "getting this under control" without widespread testing and mandatory isolation for positive tests after you end isolation.