SN$_2$ reactions proceed faster when steric hindrance is minimal, as the nucleophile attacks the electrophilic carbon directly.
Let’s evaluate steric hindrance around the carbon attached to Br in each compound:
\begin{itemize}
\item III: (C$_6$H$_5$)$_2$C(CH$_3$)Br – A tertiary benzylic halide; highly hindered $\Rightarrow$ slowest.
\item II: (C$_6$H$_5$)$_2$CHBr – Secondary benzylic halide; still hindered.
\item I: C$_6$H$_5$CH(CH$_3$)Br – Secondary benzylic halide; less hindered than II.
\item IV: C$_6$H$_5$CH$_2$Br – Primary benzylic halide; least hindered $\Rightarrow$ fastest.
\end{itemize}
Hence, SN$_2$ reactivity order: III< II< I< IV.