Long-span bridges—cable-stayed, suspension, and arch structures with main spans exceeding 300 m—are inherently susceptible to wind-induced aeroelastic instabilities, of which flutter, vortex-induced vibration (VIV), and torsional divergence represent the most structurally critical failure modes. The catastrophic collapse of the Tacoma Narrows Bridge in 1940 established the definitive precedent for aeroelastic failure of bridge decks and initiated seven decades of research into the coupled fluid-structure interaction (FSI) phenomena governing bridge aerodynamics. This paper presents a comprehensive analytical and computational aeroelastic instability analysis framework for long-span bridges under turbulent wind, integrating Scanlan's flutter derivative theory, the quasi-steady buffeting theory of Davenport and Scanlan, stochastic simulation of turbulent wind fields using the spectral representation method, and a geometrically nonlinear time-domain finite element formulation for the bridge structural system. The framework is applied to a parametric study of bridges spanning 400–900 m, covering streamlined box girder, open truss, and plate girder deck cross-sections. The critical flutter wind speed U_F is determined by eigenvalue analysis of the state-space aeroelastic system incorporating frequency-dependent flutter derivatives H₁*–H₄* and A₁*–A₄* calibrated against wind tunnel measurements. Buffeting response is computed by random vibration analysis using the Davenport-Holmes buffeting force spectrum and frequency-domain mechanical admittance functions, with root mean square (RMS) and peak displacements evaluated at multiple wind return periods. A tuned mass damper (TMD) optimisation study demonstrates that a 2% mass ratio TMD reduces RMS buffeting displacement by 42–55%