The first and second stochastic moments of the motion of a suspension bridge under the excitation of a natural wind are evaluated theoretically on the assumptions that typical scale of turbulence in the wind flow is much larger than the lateral dimensions of the bridge, and that the turbulence field is convected at a high velocity relative to the bridge. It is shown that the turbulence changes both the self-excited loads and the buffeting loads; therefore, the two types of loads are statistically correlated. This correlation generally results in a non-zero average response, even though the turbulent fluctuation has a zero mean. Furthermore, the presence of turbulence changes the motion stability boundaries, particularly the second moment stability boundary, and the mean square response can increase greatly when the stability boundary is approached.