  The Tur\'{a}n number of a graph $H$, $\text{ex}(n,H)$, is the maximum number
of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a
vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension
$\mathcal{F}+v$ of $\mathcal{F}$ is the graph obtained by connecting the vertex
$v$ to all vertices of $F$ for each $F\in \mathcal{F}$. For two integers
$k\ge1$ and $r\ge2$, let $H_i$ be a graph containing a critical edge with
chromatic number $r$ for any $i\in\{1,\ldots,k\}$, and let
$H=\{H_1,\ldots,H_k\}+v$. In this paper, we determine $\text{ex}(n, H)$ and
characterize all the extremal graphs for sufficiently large $n$. This
generalizes a result of Chen, Gould, Pfender and Wei on intersecting cliques.
We also obtain a stability theorem for $H$, extending a result of Roberts and
Scott on graphs containing a critical edge.
