The rental market in Manhattan simply isn't comparable to the rest of the US. The average cost of a 3bdr apartment in the center of Berlin is, from casual Google research, almost exactly the same in USD as a 3bdr apartment in the middle of Chicago. Chicago, in turn, is more expensive than Cleveland, Milwaukee, Minneapolis, Columbus, Omaha, &c &c.
Doesn't a three bedroom apartment actually have four rooms, three of them bedrooms? In Berlin "Dreizimmerwohnung" means that there are three rooms, period. So maybe your calculation is off by one.
