Describing the space-time evolution of urban population is a fundamental challenge in the science of cities, yet a complete theoretical treatment of the underlying dynamics is still missing. Here, we first reconstruct the evolution of London (UK) over 180 years and show that urban growth consists of an initial phase of diffusion-limited growth, followed by the development of the railway transport network and a consequential shift from central to suburban living. Such dynamics-which are analogous to angiogenesis in biological systems-can be described by a minimalist reaction-diffusion model coupled with economic constraints and an adaptive transport network. We then test the generality of our approach by reproducing the evolution of Sydney, Australia, from 1851 to 2011. We show that the rail system coevolves with urban population, displaying hierarchical characteristics that remain constant over time unless large-scale interventions are put in place to alter the modes of transport. These results demonstrate that transport schemes are first-order controls of long-term urbanization patterns and efforts aimed at creating more sustainable and healthier cities require careful consideration of population-transport feedbacks.