  Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${\rm
reg}(I^{(s)}) = {\rm reg}(I^s)$$ for $s = 2,3$, where $I^{(s)}$ is the $s$-th
symbolic power of $I$. As a consequence, we prove the following bounds
\begin{align*} {\rm reg} I^{s} & \le {\rm reg} I + 2s - 2, \text{ for } s =
2,3, {\rm reg} I^{(s)} & \le {\rm reg} I + 2s - 2,\text{ for } s = 2,3,4.
\end{align*}
