  Given a finite rank free group $\mathbb{F}$ of $\mathsf{rank}(\mathbb{F})\geq
3$, we show that the mapping torus of $\phi$ is (strongly) relatively
hyperbolic if $\phi$ is exponentially growing. We combine our result with the
work of Button-Kropholler to answer a question asked by Minasyan-Osin regarding
the acylindrical hyperbolicity of such free-by-cyclic extensions. As an
application we construct new examples of free-by-free hyperbolic extensions
where the elements of the quotient group are not necessarily fully irreducible.
We also give a new proof of the Bridson-Groves quadratic isoperimetric
inequality theorem.
