In inner cities in the US there is more poverty (as a proportion of the population) than in mid-sized cities or small towns.
Do we have a definitive answer as to why this is?
Urbanisation is a phenomenon with many effects, which can change depending of the government structure, the market structure, and the level of solidarity between citizens. In the two sections bellow, I'll try to summarize the impacts of urbanisation on rural and urban development.
Urban growth and rural populations
On the short-run, urbanization might have some positive effects for rural residents. Indeed, urban growth can provide new opportunities for rural workers (who can now find jobs in cities) and increase the demand for some products (like food-supplies) from rural areas. However, the effect of urbanisation change on the long-run and can become really negative for rural residents. Farms (and other firms centered around ressource exploitation) start facing labor shortage as young people go into cities to find work. Eventually, this shortage can lead to economic stagnation, and produce a cycle of long-term unemployment where specialized workers can't find the ressources to start new businesses. This situation can be worsened by government disengagement (ex: government refuse to provide public services to zones with decreasing population), the lack of popular organisation (you need a strong solidarity to generate durable solutions to those kind of problems), discouragement (stagnation and unemployment can lead to discouragement and depression), and globalisation (when there's no regulation on international trade, regional development can be very difficult because small firms face global competition).
Urban growth and urban populations
Concerning the urban population, urban development is largely negative. Often, urban development leads to increase in rent price, gentrification (the destruction of affordable housing to build new, costly, buildings), atomisation (loneliness), and pollution. It's those effects that lead to greater inequality. However, those effects are not determined by any economic "laws" and can be changed by regulations or citizens interventions. For exemple, laws can be enacted to conter speculation, gentrification, and pollution. Institutional structures can facilitate popular organisation, and provide a source of socialisation in our post-modern society. Urbanisation is in itself neutral - it's consequences on society is a collective choice. If this interest you, I would suggest the following readings:
Urbanization Without Cities (Bookchin)
The Death and Life of Great American Cities (Jane Jacobs)
The Next American Metropolis: Ecology, Community, and the American Dream (Calthorpe)
The denser the population, the higher the local demand for any good, and under limited supply this means higher prices. At the same time, with a surplus of labor, there is no pressure to increase wages and in fact a downward pressure on them. The natural consequence is an easily observable wealth gap that gets worse over time without government intervention.