  In this paper, we initiate the study of the extremal exponent for the
$1$-subdivisions of non-bipartite non-complete graphs. Let $K_{C_k,1}$ be the
graph by connecting an additional vertex to every vertex of the cycle $C_k$,
$k\geq 4$. We show that for the $1$-subdivision of the tripartite graph
$K_{C_k,1}^{\text{sub}}$, its Tur\'an number is $\Theta(n^{4/3})$. Then we
obtain the upper bound for the extremal number of a family of graphs consisting
of (possibly degenerate) $1$-subdivisions of certain tripartite graphs. We also
obtain the same upper bound for the 1-subdivision of $(K_{s+1,t}^+)^{sub}$,
where $K_{s+1,t}^+$ is a graph obtained by joining $K_{s,t}$ with one new
vertex $r$ by means of connecting $r$ with one vertex in the $s$-part and all
the vertices in the $t$-part.

Combinatorics