The increasing capabilities of Digital Twins (DTs) in the context of the Internet of Things (IoT) and Industrial IoT (IIoT) call for seamless integration with simulation platforms to support system design, validation, and real-time operation. This paper introduces the concept, design, and experimental evaluation of the DT Simulation Bridge - a software framework that enables diverse interaction patterns between active DTs and simulation environments. The framework supports both the DT development lifecycle and the incorporation of simulations during active operation. Through bidirectional data exchange, simulations can update DT models dynamically, while DTs provide real-time feedback to adapt simulation parameters. We describe the architectural design and core software components that ensure flexible interoperability and scalable deployment. Experimental results show that the DT Simulation Bridge enhances design agility, facilitates virtual commissioning, and supports live behavioral analysis under realistic conditions, demonstrating its effectiveness across a range of industrial scenarios.