Existing video object removal methods predominantly rely on diffusion models following a noise-to-data paradigm, where generation starts from uninformative Gaussian noise. This approach discards the rich structural and contextual priors present in the original input video. Consequently, such methods often lack sufficient guidance, leading to incomplete object erasure or the synthesis of implausible content that conflicts with the scene's physical logic. In this paper, we reformulate video object removal as a video-to-video translation task via a stochastic bridge model. Unlike noise-initialized methods, our framework establishes a direct stochastic path from the source video (with objects) to the target video (objects removed). This bridge formulation effectively leverages the input video as a strong structural prior, guiding the model to perform precise removal while ensuring that the filled regions are logically consistent with the surrounding environment. To address the trade-off where strong bridge priors hinder the removal of large objects, we propose a novel adaptive mask modulation strategy. This mechanism dynamically modulates input embeddings based on mask characteristics, balancing background fidelity with generative flexibility. Extensive experiments demonstrate that our approach significantly outperforms existing methods in both visual quality and temporal consistency. The project page is https://bridgeremoval.github.io/.