Most of human history has been spent in much lower density living situations than we have now, and I'm pretty confident there is a negative correlation between housing density and fertility.
there is no need for each home to have their own garden. but a multi story building with a dozen or more families that can share one.
take a look at some cities in germany. stuttgart, ruhrgebiet, and many others. they are actually a lot of small cities close together with lots of green space in between. each has their own shops, offices, schools, etc within walking distance and an extensive public transport network to connect them.