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In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by $g_1,\cdots ,g_k$ with order $n_1,\cdots ,n_k\in \{1,2,\cdots \}\cup \left\{\infty \right\}$, then $${\beta }_1^{\left(2\right)}\left(G\right)\le k-1-\sum _{i=1}^k\frac{1}{n_i},$$ where ${\beta }_1^{\left(2\right)}\left(G\right)$ denotes the first ${\ell }^2$-Betti number of $G$. We also show that any $k$-generated group with ${\beta }_1^{\left(2\right)}\left(G\right)\ge k-1-\epsilon$ must have girth greater than or equal $1/\epsilon$.