In general the answer is **no**.

This kind of question is studied in more generality in the paper by Mella and Polastri "Equivalent birational embeddings", Bull Lond. Math. Soc. **41** (2009) 89-93, [http://arxiv.org/abs/0906.4858.][1]

They prove that two birational embeddings of $X$ in $\mathbf{P}^n$ are equivalent up to Cremona transformations of $\mathbf{P}^n$ as long as $n\geq \dim(X)+2$. For instance, any rational variety of codimension at least $2$ in $\mathbb{P}^n$ is Cremona equivalent to a linear space.

The case $n=2$ and $\dim(X)=1$ is outside this range, and indeed there are examples of birational plane curves that are not equivalent up to Cremona transformations, hence not equivalent under the action of $\textrm{Bir}(\mathbb{P}^2)$, since $\textrm{Bir}(\mathbb{P}^2)$ is generated by Cremona transformations.

The following example is given by Mella and Polastri in the last section of their paper. Take a general projection of a curve of bidegree $(1,d)$ on a quadric surface to $\mathbb{P}^2$. This is a plane rational curve $C$ of degree $d$ with only ordinary double points, hence there is a birational isomorphism $C \dashrightarrow L$, where $L$ is a line. However, one proves that if $d \geq 6$ then $C$ is **not** Cremona equivalent to $L$. 

The proof is based on the following 

> **Lemma.** Let $X \subset \mathbb{P}^n$ be a rational variety of codimension $1$ and degree $d>1$. If $X$ is Cremona equivalent to a hyperplane, then the singularities of the pair $(\mathbb{P}^n, \frac{n+1}{d} X)$ are not canonical. 

  [1]: http://arxiv.org/abs/0906.4858