  We show that every self conformal measure with respect to a $C^2
(\mathbb{R})$ IFS $\Phi$ has polynomial Fourier decay under some mild and
natural non-linearity conditions. In particular, every such measure has
polynomial decay if $\Phi$ is $C^\omega (\mathbb{R})$ and contains a non-affine
map.
  A key ingredient in our argument is a cocycle version of Dolgopyat's method,
that does not require the cylinder covering of the attractor to be a Markov
partition. It is used to obtain spectral gap-type estimates for the transfer
operator, which in turn imply a renewal theorem with an exponential error term
in the spirit of Li (2022).
