  In many proofs concerning extremal parameters of Berge hypergraphs one starts
with analyzing that part of that shadow graph which is contained in many
hyperedges. Capturing this phenomenon we introduce two new types of
hypergraphs. A hypergraph $\mathcal{H}$ is a $t$-heavy copy of a graph $F$ if
there is a copy of $F$ on its vertex set such that each edge of $F$ is
contained in at least $t$ hyperedges of $\mathcal{H}$. $\mathcal{H}$ is a
$t$-wise Berge copy of $F$ if additionally for distinct edges of $F$ those $t$
hyperedges are distinct.
  We extend known upper bounds on the Tur\'an number of Berge hypergraphs to
the $t$-wise Berge hypergraphs case. We asymptotically determine the Tur\'an
number of $t$-heavy and $t$-wise Berge copies of long paths and cycles and
exactly determine the Tur\'an number of $t$-heavy and $t$-wise Berge copies of
cliques.
  In the case of 3-uniform hypergraphs, we consider the problem in more details
and obtain additional results.
