  For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an
$R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs
to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a
Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to
E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this
paper, we define a variant of Tur\'an number in hypergraphs, namely the cover
Tur\'an number, denoted as $\hat{ex}_R(n, G)$, as the maximum number of edges
in the shadow graph of a Berge-$G$ free $R$-graph on $n$ vertices. We show a
general upper bound on the cover Tur\'an number of graphs and determine the
cover Tur\'an density of all graphs when the uniformity of the host hypergraph
equals to $3$.
