You're probably just thinking of Times Square. The vast majority of people that live even in Manhattan do not live in horribly suffocating/overcrowded neighborhoods. To give you one example, I live in Stuyvesant Town, which has a population density of 200k/mi^2 (which is greater than the overall population density of Manhattan at 170k/mi^2), and yet Stuyvesant Town feels positively residential. It's full of parks, playgrounds, trees, green spaces, walking paths, etc., and it certainly never feels overcrowded. If anything it feels like a respite from the parts of Manhattan that do get super-crowded, which are inevitably the commercial/tourististic hubs.

It takes population densities of greater than 1M/mi^2 to truly feel overcrowded, and even in NYC you only see such densities in localized incredibly high-traffic spots, like Times Square and Herald Square. The vast majority of Manhattan is never so crowded as to feel suffocating.