The dynamics of a structurally non-linear two-dimensional airfoil in turbulent flow is investigated numerically using a Monte Carlo approach. Both the longitudinal and vertical components of turbulence, corresponding to parametric (multiplicative) and external (additive) excitation, respectively, are modelled. The properties of the airfoil are chosen such that the underlying non-excited, deterministic system exhibits binary flutter; the loss of stability of the equilibrium point due to flutter then leads to a limit cycle oscillation (LCO) via a supercritical Hopf bifurcation. For the random system, the results are examined in terms of the probability structure of the response and the largest Lyapunov exponent. The airfoil response is interpreted from the point of view of the concepts of D- and P-bifurcations, as defined in random bifurcation theory. It is found that the bifurcation is characterized by a change in shape of the response probability structure, while no discontinuity in the variation of the largest Lyapunov exponent with airspeed is observed. In this sense, the trivial bifurcation obtained for the deterministic airfoil, where the D- and P-bifurcations coincide, appears only as a P-bifurcation for the random case. At low levels of turbulence intensity, the Gaussian-like bell-shaped bi-dimensional PDF bifurcates into a crater shape; this is interpreted as a random fixed point bifurcating into a random LCO. At higher levels of turbulence intensity, the post-bifurcation PDF loses its underlying deterministic LCO structure. The crater is transformed into a two-peaked shape, with a saddle at the origin. From a more universal point of view, the robustness of the random bifurcation scenario is critiqued in light of the relative importance of the two components of turbulent excitation.