Polynomial regression fits a non-linear model to the data. But as a statistical estimation problem it's still linear in the sense that the regression function $h\left(\Theta, X\right)$ is linear in the unknown parameters $\Theta$.
When we use polynomial regression we actually give our linear model additional features like $X^2$ or $XY$. But with the same success you can give your model features like $\log\left(X\right)$ or $\exp\left(X\right)$, and after that apply least squares. So you can fit any kind of curvature to your data.
My question is: Why does non-linear regression assume a more general hypothesis space of functions - one that encompasses the hypothesis space of functions that you can get with linear regression? I mean why do we think that non-linear regression can fit more types of curvatures to the data than linear regression if linear regression itself (e.g. with polynomial or logarithmic features) can fit any curvature to the data?