Here is an example. Consider a map $f:\mathbb R^2\to\mathbb R^2$ given by $(r,\varphi)\mapsto (r,\varphi+\sin(1/r))$ is polar coordinates $(r,\varphi)$, $0<r\le 1/\pi$. For $r>1/\pi$, let the map be identity map, then close up the plane to make a compact manifold.

This map has no conjugate $C^1$ diffeomorphism. Indeed, we may assume that 0 is mapped to itself by the conjugation. Then the circles centered at the origin are mapped to Jordan curves winding around the origin. There are arbitrarily small circles made of fixed points, hence the derivative of the diffeomorphism at the origin is the identity. On the other hand, there are arbitrary small circles on which our map is periodic with rotation number, say, 1/10. It is easy to see that this cannot happen on a Jordan curve winding around the origin, unless some point changes is angular coordinate by $2\pi/10$. This contradicts the fact that the derivative at 0 must be the identity.

Here is an example. Consider a map $f:\mathbb R^2\to\mathbb R^2$ given by $(r,\varphi)\mapsto (r,\varphi+\sin(1/r))$ in polar coordinates $(r,\varphi)$, $0<r\le 1/\pi$. For $r>1/\pi$, let $f$ be identity, then close up the plane to make a compact manifold.

This map has zero topological entropy but has no conjugate $C^1$ diffeomorphism. Indeed, we may assume that 0 is mapped to itself by the conjugation. Then the circles centered at the origin are mapped to Jordan curves winding around the origin. There are arbitrarily small circles made of fixed points, hence the derivative of the diffeomorphism at the origin is the identity. On the other hand, there are arbitrary small circles on which our map is periodic with rotation number, say, 1/10. It is easy to see that this cannot happen on a Jordan curve winding around the origin, unless some point changes is angular coordinate by $2\pi/10$. This contradicts the fact that the derivative at 0 is the identity.