Interruptions, a fundamental component of human communication, can enhance the dynamism and effectiveness of conversations, but only when effectively managed by all parties involved. Despite advancements in robotic systems, state-of-the-art systems still have limited capabilities in handling user-initiated interruptions in real-time. Prior research has primarily focused on post hoc analysis of interruptions. To address this gap, we present a system that detects user-initiated interruptions and manages them in real-time based on the interrupter's intent (i.e., cooperative agreement, cooperative assistance, cooperative clarification, or disruptive interruption). The system was designed based on interaction patterns identified from human-human interaction data. We integrated our system into an LLM-powered social robot and validated its effectiveness through a timed decision-making task and a contentious discussion task with 21 participants. Our system successfully handled 93.69% (n=104/111) of user-initiated interruptions. We discuss our learnings and their implications for designing interruption-handling behaviors in conversational robots.