jmc

We don't know $g(x),$ so we don't have an expression we can simply stick $4$ in to get an answer. We do, however, know that $g(f(x))=5-4x.$ So, if we can figure out what to put into $f(x)$ such that $4$ is output, we can use our expression for $g(f(x))$ to find $g(4).$

Since $f(x)=2x-3,$ the value of $x$ such that $f(x)=4$ is the solution to the equation $2x-3=4,$ which is $x=7/2.$ So, we have $f(7/2)=4.$ Therefore, if we let $x=7/2$ in $g(f(x))=5-4x,$ we have $$g(f(7/2))=5-4\cdot \frac{7}{2}⟹g(4)=5-14=\boxed{-9}.$$