This paper investigates the global dynamic behavior and bifurcations of a classical nonlinear transmission SIR epidemic model with discontinuous threshold strategy. Different from previous results, we not only consider the general nonlinear transmission, but also adopt the discontinuity control. First, the positivity and boundedness of the model are given. Second, by employing Lyapunov LaSalle approach and using Green Theorem, we perform the globally stable the three types of equilibria of the system. We analytically show the orbit can tend to the disease-free equilibrium point, the endemic equilibrium point or the sliding equilibrium point in discontinuous surfaces of the system. In addition, we also analyze the sliding bifurcations of the model when consider the special transmission. Finally, some numerical simulations are worked out to confirm the results obtained in this paper.