I have an algebraic group $G$ acting on an affine variety $X$, the orbit $O(m)$ of an element $m \in X$, and an affine curve $C$ contained in the Zariski closure $\overline{O(m)}$ of $O(m)$, such that $m \in C$.

If we define $C^\prime = C \cap O(m) $ then it's not hard to see that $C^\prime$ is open in $C$. And therefore $C^\prime$ is again affine (follows for example from an exercise in Hartshorne).

Let $p: G \to O(m)$ defined by $p(g) = g \cdot m$ be the orbit map. I want to prove that there is an affine subset $G^\prime$ of $G$ such that $p(G^\prime)$ is dense in $C^\prime$.

Is the preimage of $X^\prime$ affine? If so why? If not, how do I construct such a $G^\prime$?

For reference: The statement that such an $G^\prime$ exists can be found in the article A characterization of orbit closure and applications (MR0944153) on the first page of the proof of Theorem 1.2. Clearly he has more assumptions on $X$ and $G$, but I do not see anything else that could be useful here.