  For a graph $H$ and a $k$-chromatic graph $F,$ if the Tur\'an 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 $F$-Tur\'an-good, or
$k$-Tur\'an-good for short if $F$ is $K_k.$ In this paper, we construct some
new classes of $k$-Tur\'an-good graphs and prove that $P_4$ and $P_5$ are
$k$-Tur\'an-good for $k\ge4.$
