There’s more to it than housing density (see below). For instance, there are also differences in public transport to spread the area of viable housing. Anecdotally, high rent cities such as New York or London tend to not do a great job at providing reasonably priced and efficient transport to nearby commuter towns or suburban areas compared to many in central Europe, for instance.

On densities, cities like New York, San Francisco, and Seattle (high rents) are comparable or higher in density than larger European cities with the exception of Paris (and I may have missed another). London, Berlin, or Madrid are noticeably lower than New York or possibly the other two US cities as well.

(Naturally, the city boundaries of somewhere like London include many less-urban areas so this doesn’t quite capture exactly what either of us presumably mean, but it’s loosely demonstrative.)