In this paper we study the Cayley graph $\mathrm{Cay}(S_n,T)$ of the symmetric group $S_n$ generated by a set of transpositions $T$. We show that for $n\geq 5$ the Cayley graph is normal. As a corollary, we show that its automorphism group is a direct product of $S_n$ and the automorphism group of the transposition graph associated to $T$. This provides an affirmative answer to a conjecture raised by Ganesan in arXiv:1703.08109, showing that $\mathrm{Cay}(S_n,T)$ is normal if and only if the transposition graph is not $C_4$ or $K_n$.