  We estimate the maximum possible number of cliques of size $r$ in an
$n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2,
\ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on
the Tur\'an number of the complete $r$-partite hypergraphs gives the upper
bound $O\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. We improve this to
$o\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. The main tool in our proof
is the graph removal lemma. We also provide several lower bound constructions.
