Road traffic casualties represent a hidden global epidemic, demanding evidence-based interventions. This paper demonstrates a network lattice approach for identifying road segments of particular concern, based on a case study of a major city (Leeds, UK), in which 5,862 crashes of different severities were recorded over an eight-year period (2011-2018). We consider a family of Bayesian hierarchical models that include spatially structured and unstructured random effects, to capture the dependencies between the severity levels. Results highlight roads that are more prone to collisions, relative to estimated traffic volumes, in the northwest and south of city-centre. We analyse the Modifiable Areal Unit Problem (MAUP), proposing a novel procedure to investigate the presence of MAUP on a network lattice. We conclude that our methods enable a reliable estimation of road safety levels to help identify "hotspots" on the road network and to inform effective local interventions.