Two smooth tangent almost complex curves in a $4$-manifold

This follows from theorem 6.2 (and the first sentence in the proof) of Mario J. Micallef and Brian White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. (2) 141 (1995), no. 1, 35–85.

It turns out that it is nice to read books. The answer to the weaker version of my question with $O|z|^{n+1}$ term is contained on page 17 of McDuff-Salamon book [MS] (no need of Micallef-White!): https://people.math.ethz.ch/~salamon/PREPRINTS/jholsm.pdf

Proof. In the proof of Lemma 2.2.3 of [MS] one uses coordinates in $\mathbb C^2$ such that $C_1$ is given by $w=0$ and the almost complex structure $J$ along the line $(z,0)$ is the standard one. Then it is explained that the almost complex map $z\to \mathbb C^2$ corresponding to $C_2$ is given by

$$z\to (p(z)+O(|z^{n+1}|), az^n+O(|z^{n+1}|))$$

where $p(z)$ is a polynomial of order at most $n$, $a\ne 0$. In our case of course $p'(0)\ne 0$. It is now clear that in these coordinates $C_2$ is as need. QED.

Comment. The above proof is elementary and does not use Micallef-White. Similarly to Micallef-White's, statement it can be used to answer the original question with a $C^1$-smooth change of coordinates (instead of $C^{\infty}$). Indeed, after a smooth reparameterization in $z$ and scaling in $w$ the above map for $C_2$ looks as $$z\to (z, z^n+O(|z^{n+1}|)).$$ Denote the second term by $f(z)$. Then the map $(z,w)\to (z,w-(f(z)-z^n)/z^n)$ is $C^1$ and it sends the couple $C_1,C_2$ to the couple $(w=0, w=z^n)$.

I wonder still if one can make this last change of coordinates $C^{\infty}$...