For a graph $H$ and a $k$-chromatic graph $F,$ if the Turán graph $T_{k-1}(n)$ has the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough), then $H$ is called $F$-Turán-good, or $k$-Turán-good for short if $F$ is $K_k.$ In this paper, we construct some new classes of $k$-Turán-good graphs and prove that $P_4$ and $P_5$ are $k$-Turán-good for $k\ge4.$