Accurate multi-step traffic forecasting in urban networks is essential for intelligent transportation systems, yet a key challenge remains: standard models often treat road segments independently, ignoring the critical spatial dependencies imposed by the road network's topology. To address this, we introduce a novel forecasting framework that integrates network structure directly into a probabilistic time series model. Our method develops a Bayesian Structural Time Series (BSTS) model for each key road segment, incorporating local trend, daily seasonality, and dynamic regressors. The central innovation is the application of a graph Laplacian prior on the posterior distributions of the contemporaneous coefficients across the network. This regularization, informed by the actual connectivity graph, facilitates information sharing between neighboring segments, thereby penalizing implausible spatial discontinuities in the learned parameters. Inference is performed via Markov Chain Monte Carlo (MCMC) sampling. Applied to high-frequency data from a 50-node subnetwork of a major European city, our graph-regularized model reduced the mean absolute percentage error (MAPE) for 60-minute forecasts by 18% on average compared to independent BSTS models and significantly outperformed vector autoregression (VAR) and LSTM benchmarks, especially during non-recurrent congestion. This demonstrates that formally incorporating network science principles via a graph-Laplacian prior into a state-space statistical framework yields substantially improved and spatially coherent forecasts, a methodology generalizable to other networked time series problems such as in economics or epidemiology.