Large, irregular stress fluctuations under a constant applied shear rate are observed during the flow of dilute worm-like micellar solutions even at low Reynolds numbers. Statistical properties of such fluctuations showing signatures of chaos and elastic turbulence have been studied extensively. Although the mechanisms like boundary slippage, dynamics of shear band interface, time-dependent secondary flows, and inertio-elastic effects are conceived as the possible factors for such striking flow properties, their contributions in different non-linear flow regimes remain poorly understood. Here, we study the Taylor–Couette flow of a well-characterized aqueous worm-like micellar system formed by 2 wt. % cetyltrimethylammonium tosylate and 100 mM sodium chloride (2 wt. % CTAT + 100 mM NaCl). For a fixed applied shear-rate just beyond the onset of shear-thinning, high-speed optical imaging in the flow-gradient plane reveals spatiotemporally varying velocity gradients in the system. In this regime, the magnitude of stress fluctuations remains insignificant. However, the fluctuation becomes substantial beyond a critical shear rate deep inside the non-linear regime of the flow curve when significant free-surface undulations, sustained stick-slip, and elastic recoil events are observed. Imaging in the flow-vorticity and the gradient-vorticity plane indicates that such dynamics are primarily driven by the elasticity-induced turbulent flows in the system. Furthermore, in this regime, we find that the characteristic persistent time of stress fluctuations matches well with the time scales of the stick-slip events, as well as the micellar breaking time, indicating a possible connection between the striking stress dynamics and the micellar kinetics.