In this study we apply an extended version of the SEsM to find exact solutions of the conformable time–fractional KdV–type equation, which describes higher dispersive and memory effects appearing in shallow water dynamics. In accordance with the extended SEsM we construct the general solution of the studied equation as composite function of two simpler functions with two independent variables. The composing functions include the solutions of two simple equations with different independent variables. We chose the simple equations to be ordinary differential equations (ODEs) of second order. Several solutions are derived in dependence on numerical values of the coefficients of the used simple equations.