Emerging satellite Internet constellations such as SpaceX's Starlink will deploy thousands of broadband satellites and construct Low-Earth Orbit (LEO) satellite networks (LSNs) in space, significantly expanding the boundaries of today's terrestrial Internet. However, due to the unique global LEO dynamics, satellite routers will inevitably pass through uncontrolled areas, suffering from security threats. It should be important for satellite network operators (SNOs) to enable verifiable riskavoidance routing to identify path anomalies. In this paper, we present STARVERI, a novel network path verification framework tailored for emerging LSNs. STARVERI addresses the limitations of existing crypto-based and delay-based verification approaches and accomplishes efficient and accurate path verification by: (i) adopting a dynamic relay selection mechanism deployed in SNO's operation center to judiciously select verifiable relays for each communication pair over LSNs; and (ii) incorporating a lightweight path verification algorithm to dynamically verify each segment path split by distributed relays. We build an LSN simulator based on real constellation information and the results demonstrate that STARVERI can significantly improve the path verification accuracy and achieve lower router overhead compared with existing approaches.