The rapid aging of urban transportation infrastructure presents a significant safety risk and financial burden for municipal authorities. Traditional Structural Health Monitoring (SHM) techniques often rely on periodic manual inspections or sensor-heavy data streams that lack the context of physical structural laws. This paper introduces a "Physics-Informed Neural Network" (PINN) framework that integrates real-time IoT sensor data with fundamental structural mechanics (Euler-Bernoulli beam theory and Navier equations) to monitor bridge integrity. Unlike standard "black-box" AI models, our PINN approach ensures that predictions adhere to the laws of physics, such as mass conservation and material stiffness constraints. Using a simulated multi-span highway bridge, we demonstrate the model’s ability to detect sub-surface fatigue cracking and load-bearing anomalies with 95% accuracy, even with sparse sensor coverage. The framework allows for the creation of a "Dynamic Digital Twin" that evolves with the structure's wear, enabling a shift from reactive to proactive maintenance. Our results show that PINN-driven monitoring can extend the service life of aging bridges by up to 15 years while reducing inspection costs by 40%.