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.
That's not really answering your question, but for a gigantor metropolis berlin is a lot cheaper than manhattan...