We consider the Cauchy problem for the damped wave equation with absorption $$u_{tt}-\Delta u +u_t +|u|^{\rho-1}u =0, \quad (t,x) \in \mathbf{R}_+ \times \mathbf{R}^N, \qquad(*)$$ with $N=3,4$ . The behavior of $u$ as $t\to \infty$ is expected to be the Gauss kernel in the supercritical case $\rho > \rho_c(N):=1+2/N$ . In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175--197) for $\rho>1+\frac{4}{N}(N=1,2,3)$ , Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for $\rho>\rho_c(N)(N=1)$ and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598--610) for $\rho_c(N)<\rho\le 1+\frac{4}{N}(N=1,2)$ and $\rho_c(N)<\rho< 1+\frac{3}{N}(N=3)$ . Developing their result, we will show the behavior of solutions for $\rho_c(N)<\rho\le 1+\frac{4}{N}(N=3)$ , $\rho_c(N)<\rho < 1+\frac{4}{N}(N=4)$ . For the proof, both the weighted $L^2$ -energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464--489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for $N=5$ , because the semilinear term is not in $C^2$ and the second derivatives are necessary when the explicit formula of solutions is estimated.