Imagine in $C^r$-Topology we have that $f,g \in \text{Diff}^{r}(M)$ and $\phi(g)$ is a continuous function which $g^n\phi(g)(x)=\phi(g)f^n(x)$ I have the following relation:
\begin{align} d(f^n(x),g^n\phi(g)(x))=d(f^n(x),\phi(g)f^n(x)) \leq d(id , \phi(g)) \end{align} I don't understand the reason of the last inequality. Is this because of the following? :
\begin{align} d(f^n(x) , \phi(g)f^n(x)) \leq d(f^n , \phi(g)f^n) \end{align}
It seems from your notation that you want $f$ to be the center of a $C^r$-neighborhood, $g$ to be in said neighborhood, and $\phi$ to be a conjugacy between $f$ and $g$. Though this question doesn't seem to do much with the $C^r$ topology per se. Rather, it's about the $C^0$ distance between continuous functions.
First, second and fourth instances of $d$ signify the distance function of $M$, whereas the third and fifth instances signify the distance function that is associated to the $C^0$-topology on the space of continuous self-maps of $M$. To be more explicit let's denote the former by $d_M$ and the latter by $d_{C^0}$. Then by the definition of $d_{C^0}$ we have for $\alpha,\beta\in C^0(M,M)$:
$$d_{C^0}(\alpha,\beta)=\sup_{p\in M} d_M(\alpha(p),\beta(p)),$$
and one can put $p=f^n(x)$ as $f:M\to M$ (and $id=\operatorname{id}_M:M\to M$).