This paper presents the design, analysis, and performance evaluation of an omnidirectional transformable wheel-leg robot called OmniWheg. We design a novel mechanism consisting of a separable Omni-wheel and 4-bar linkages, allowing the robot to transform between Omni-wheeled and legged modes smoothly. On wheeled mode, the robot can move in all directions and efficiently adjust the relative position of its wheels, while it can overcome common obstacles in legged mode, such as stairs and steps. Unlike other articles studying whegs, this implementation with omnidirectional wheels allows the correction of misalignments between right and left wheels before traversing obstacles, which effectively improves the success rate and simplifies the preparation process before the wheel-leg transformation. We describe the design concept, mechanism, and the dynamic characteristic of the wheel-leg structure. We then evaluate its performance in various scenarios, including passing obstacles, climbing steps of different heights, and turning/moving omnidirectionally. Our results confirm that this mobile platform can overcome common indoor obstacles and move flexibly on the flat ground with the new transformable wheel-leg mechanism, while keeping a high degree of stability.