Global climate change leads to frequent extreme snowfall weather, which has a significant impact on the safety and operating efficiency of urban public transportation. In order to cope with the adverse effects of extreme weather, governments should vigorously develop sustainable transportation. Since urban public transportation is a critical component of building a sustainable city, traffic management departments should quantitatively analyze the performance changes of the urban public transportation network under extreme weather conditions. Therefore, fully considering the comprehensive effects of network performance and topology to improve the robustness of urban public transportation systems requires more attention. The urban public transport network with high robustness can achieve fewer recovery costs, lower additional bus scheduling costs, and achieve the sustainable development of the public transport network. Considering the impact of travelers’ travel time tolerance and in-vehicle space congestion tolerance under snowy conditions, this paper proposes a generalized-norm-based robustness evaluation model of the bus network. Example analyses are conducted using checkerboard and ring-radial topological network structures to verify the applicability of the proposed model. The results show the following: (1) In an extreme snowfall scenario, the robustness of checkerboard and ring-radiating bus networks is reduced by 38% and 39%, respectively. (2) In the checkerboard network, the central area units are always more important to the system robustness than the peripheral units, while, in the ring-radial network, the units with higher importance are all in the ring line. (3) The failure of Ring Line 5 has a great impact on both the checkerboard and ring-radial networks, causing the system robustness to decrease by 43% and 50%, respectively.