Distributed optical fiber sensors (DOFSs) have faced the challenge of measuring transverse stress along the fiber and the current main approach has been based on polarization coupling effect in polarization-maintaining fibers (PMFs), which has short sensing length and high dependence on direction of exerted stress. Instead, here we propose a novel distributed transverse stress sensor (DTSS) based on coupling effect between linearly-polarized (LP) modes in weakly-coupled few-mode-fibers (FMFs). In this scheme, multiple LP modes could be considered as independent spatial channels without stress perturbation because of ultralow inherent modal crosstalk, while quantifiable and spatially-resolvable mode coupling for a probe signal will occur under transverse stress satisfying phase-matching conditions. A proof-of-concept DTSS system is verified based on weakly-coupled two-mode fibers and mode-selective couplers for mode conversion. Moreover, we show that the scheme is little affected by mild common parameters including temperature, strain, twist, direction of stress, or state-of-polarization (SOP), which is crucial for accurate stress analysis under complex environmental conditions. The proposed DTSS scheme has simple structure, high flexibility for different sensing ranges and resolutions, and high collaborating capability with other sensing mechanisms.