If fixed-point iteration has linear convergence, how can Newton's Method have quadratic convergence?

Question

Newton's Method for finding the roots of a function can be considered a type of fixed point iteration of $g(x) = x - \frac{f(x)}{f'(x)}$, since $f(k) = 0 \rightarrow g(k) = k$. But it is well-known that fixed point iteration has linear convergence while Newton's Method has quadratic convergence, given some assumptions on $f$ and $g$ such that $f'(k) \neq 0$. How is this possible? Shouldn't Newton's Method have the same rate of convergence in this case as fixed point iteration?