There is some truth to the idea though. Imagine living in a big city and you want to have kids, having kids means you need extra living space/hospitals/schooling/cars etc. You can't just easily go buy a townhouse for cheap. Neighborhood crime also becomes a concern. So it means disrupting your life, socially, culturally, and work wise and moving to the suburbs or a small town.

I doubt this is just dense urbanization<>low rates, it's mostly just modern urbanization where everyone wants to live in the city but governments/NIMBYs refuse to build any housing parents or retail/services businesses supporting children can afford. So costs are artificially exorbitant. Which brings down the birth rates, so immigration is increased who also want to live in cities, and need housing and want kids...etc. So there is both artificial downward pressures on birth rates in (modern) cities and naturally via the general chaos of cities.