In this work, we investigate the time discretization approximation of nonlinear evolution equations governed by semibounded or dissipative operators in a general normed space. Some properties of the semibounded and dissipative operators are first explored by employing the semi-inner product of the normed space. These properties play crucial role in the strong stability and long time convergence analysis of two classes of high order numerical methods, diagonally implicit Runge-Kutta (DIRK) and stepsize-dependent linear multistep (SDLM) methods in the normed space. They are showed to preserve the stability and contractivity of the nonlinear coercive and dissipative systems for sufficiently small time stepsize, respectively. The linear growth error estimates of high order DIRK and SDLM methods are therefore derived for dissipative problems. We present numerical results to support the theoretical findings on the linear diffusion-convection equations and nonlinear porous medium vector field.