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The paper introduces a novel polynomial compilation technique for the sound and complete removal of conditional effects in classical planning problems. Similar to Nebel's polynomial compilation of conditional effects, our solution also decomposes each action with conditional effects into several simpler actions. However, it does so more effectively by exploiting the actual structure of the given conditional effects. We characterise such a structure using a directed graph and leverage it to significantly reduce the number of additional atoms required, thereby shortening the size of valid plans. Our experimental analysis indicates that this approach enables the effective use of polynomial compilations, offering benefits in terms of modularity and reusability of existing planners. It also demonstrates that a compilation-based approach can be more efficient, either independently or in synergy with state-of-the-art optimal planners that directly support conditional effects.