How do you explain that in $y=x^2$, y is proportional to the square of x? My understanding is that all proportional relationships are linear relationships. If this is indeed the case, how is it that we can also say that in a non linear equation like $y = x^2$, y is proportional to $x^2$?

$y$ is not proportional to $x$, beacause there is a quadratic relation between $x$ and $y$, but $y$ is proportional to $x^2$ as $y=\alpha x^2$ with $\alpha=1$.
