Relations between matrix norm and determinant

Consider the matrices of the form $\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right)$, with $x\in(0,+\infty)$. All of them have determinant equal to $1$, but their norms are arbitrarily large. Does this answer your question?

I don't see any way to obtain an upper bound of $\|Ax\|$ using the determinant, because the norm of a matrix can remain unchanged when the determinant approaches zero. E.g. when $x=(1,0,\ldots,0)$ and $A=\operatorname{diag}(1,t,\ldots,t)$, we have $\|Ax\|_2=1$ but $\det A\to0$ as $t\to0$.