Text-guided multispectral object detection uses text semantics to guide semantic-aware cross-modal interaction between RGB and IR for more robust perception. However, notable limitations remain: (1) existing methods often use text only as an auxiliary semantic enhancement signal, without exploiting its guiding role to bridge the inherent granularity asymmetry between RGB and IR; and (2) conventional data-driven attention-based fusion tends to emphasize stable consensus while overlooking potentially valuable cross-modal discrepancies. To address these issues, we propose a semantic bridge fusion framework with bi-support modeling for multispectral object detection. Specifically, text is used as a shared semantic bridge to align RGB and IR responses under a unified category condition, while the recalibrated thermal semantic prior is projected onto the RGB branch for semantic-level mapping fusion. We further formulate RGB-IR interaction evidence into the regular consensus support and the complementary discrepancy support that contains potentially discriminative cues, and introduce them into fusion via dynamic recalibration as a structured inductive bias. In addition, we design a bidirectional semantic alignment module for closed-loop vision-text guidance enhancement. Extensive experiments demonstrate the effectiveness of the proposed fusion framework and its superior detection performance on multispectral benchmarks. Code is available at https://github.com/zhenwang5372/Bridging-RGB-IR-Gap.