Spatio-Temporal Video Grounding requires jointly localizing target objects across both temporal and spatial dimensions based on natural language queries, posing fundamental challenges for existing Multimodal Large Language Models (MLLMs). We identify two core challenges: \textit{entangled spatio-temporal alignment}, arising from coupling two heterogeneous sub-tasks within the same autoregressive output space, and \textit{dual-domain visual token redundancy}, where target objects exhibit simultaneous temporal and spatial sparsity, rendering the overwhelming majority of visual tokens irrelevant to the grounding query. To address these, we propose \textbf{Bridge-STG}, an end-to-end framework that decouples temporal and spatial localization while maintaining semantic coherence. While decoupling is the natural solution to this entanglement, it risks creating a semantic gap between the temporal MLLM and the spatial decoder. Bridge-STG resolves this through two pivotal designs: the \textbf{Spatio-Temporal Semantic Bridging (STSB)} mechanism with Explicit Temporal Alignment (ETA) distills the MLLM's temporal reasoning context into enriched bridging queries as a robust semantic interface; and the \textbf{Query-Guided Spatial Localization (QGSL)} module leverages these queries to drive a purpose-built spatial decoder with multi-layer interactive queries and positive/negative frame sampling, jointly eliminating dual-domain visual token redundancy. Extensive experiments across multiple benchmarks demonstrate that Bridge-STG achieves state-of-the-art performance among MLLM-based methods. Bridge-STG improves average m\_vIoU from $26.4$ to $34.3$ on VidSTG and demonstrates strong cross-task transfer across various fine-grained video understanding tasks under a unified multi-task training regime.