•The time-fractional Klein–Gorden equation quadratic and cubic nonlinearities is considered and the exact solutions to this equation is obtained.•The time fractional Klein–Gordon equation is proposed.•The classification of single traveling wave solutions to the equation is gotten.•The influence of the fractional order is analyzed. In this paper, we solve the conformal time-fractional Klein–Gordon equation with high-order nonlinearities via the complete discrimination system for polynomial method, namely with the quadratic and cubic nonlinearities. Our results contain all the existing solutions, and some new exact solutions such as Jacobian elliptic function solutions, are also presented. Moreover, to ensure the existence of each solution, the concrete examples are given and the discussions of the influence of the fractional order are also presented.