Manhattan contained 854,000 housing units in 2014 [1] and 1.6 million residents between 2010 [2] and 2013 [3]. The average Manhattan unit has 1.2 to 1.3 bedrooms [4]. (For Core Manhattan, that average falls to 1.0 to 1.2.) So 1 to 1.1 million bedrooms.
If you mandated one bedroom per person you'd have to expel 31 to 38% of the population or increase the number of bedrooms by 45 to 60%.
Manhattan's 33.6 square miles [5] are populated with 70,826 people per square mile. If we go the culling-people route, that drops to 44,000 to 49,000 people per square mile. That's the population density of West New York, in New Jersey [6]. Manhattan and West New York are radically different places.
If we go the building-units route, consider that Manhattan gained 13,189 units between 2011 and 2014, i.e. 4,400 a year [1; Table 2]. To build the 500,000 to 600,000 bedrooms we need you'd need to make 415,000 to 500,000 units, i.e. a century of construction. (If you built only three bedrooms, you'd cut that time down to 167,000 to 200,000 units, i.e. a mere 38 to 45 years.) To build them in 20 years, you'd need to increase construction by 4.7 to 5.7x (90% to 2.3x if only 3BR), and that's ignoring any additional infrastructure that new population may need as well as population growth.
Cities are a product of density. If you try to make a city with suburban-sized personal spaces, you end up with neither city nor suburbia. Building up is better than sideways, always, but we shouldn't delude ourselves regarding realistic sets of trade-offs.
[1] https://www1.nyc.gov/assets/hpd/downloads/pdf/2014-HVS-initi... Page 1
[2] http://www1.nyc.gov/assets/planning/download/pdf/data-maps/n... Table 1
[3] https://en.wikipedia.org/wiki/Demographics_of_Manhattan
[4] http://furmancenter.org/files/FurmanCenter_FactBrief_RentSta... Table F
[5] https://en.wikipedia.org/wiki/Manhattan
[6] https://en.wikipedia.org/wiki/West_New_York,_New_Jersey