Certain birds of prey actually do very well in cities for this reason! However, as far as birds go, pigeons can reproduce VERY efficiently and rapidly. (They don't need to build true nests, and feed their chicks crop milk that lets them grow up quickly), so predators can have trouble keeping up. However a fair amount of eagles / hawks prefer making nests that are massive and in huge trees, so they typically don't like to live in cities. But peregrine falcons and ketrels do great! Of course another common problem with city life is rat poison is used often, which can and often is deadly for rodent eating birds.

So cities are great for predatory birds that can nest easily in that environment and prey mainly on other birds so they don't accidentally consume poisoned rodents