Let $G$ be a finite simple connected graph on the vertex set $V(G)=[d]=\{1,\dots ,d\}$ with edge set $E(G)=\{e_1,\dots ,e_n\}$. Let $\mathbb{K}[\mathbf{\text{t}}]=\mathbb{K}[t_1,\dots ,t_d]$ be the polynomial ring in d variables over a field $\mathbb{K}$. The edge ring of G is the affine semigroup ring $\mathbb{K}[G]$ generated by monomials ${\mathbf{\text{t}}}^e:=t_i{t}_j$, for $e=\{i,j\}\in E(G)$. In this paper, we will prove that, given integers $d$ and $n$, where $d\ge 7$ and $d+1\le n\le \frac{d^2-7d+24}{2}$, there exists a finite simple connected graph $G$ with $|V(G)|=d$ and $|E(G)|=n$, such that $\mathbb{K}[G]$ is non-normal and satisfies $(S_2)$-condition.