IV - Is this procedure valid? I have three variables: $y$, $x_1$ and $z$.

I'm interested in finding the effect of $x_1$ on $y$.
Unfortunately $x_1$ is endogenous so I use an instrument z.

I know that $z$ is correlated with $x_1$. The problem is, $z$ is correlated with another variable ($x_2$) that arguably has an effect on $y$. $z$ is unlikely to have a causal effect on $x_2$, but $x_2$ may still be endogenous for other reasons (e.g. reverse causality, etc.).

Will the following procedure give me a consistent estimate of $\beta_1$:

two stage least squares with $y$ regressed on $x_1$ and $x_2$, with $z$ as an instrument for $x_1$?
Actual availability of regressors may be an issue here, but if all four mentioned variables are available, the situation is as @Michael mentioned in a comment:

Since $X_2$ is correlated with $Y$, it should be included in the regression specification as a "control".  

$$Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + u$$

This is intuitive, but it also takes care of the technical aspect.
Then since $X_2$ is endogenous, i.e we have ${\rm Cov}(X_2,u) \neq 0$, but we have an instrument that is not correlated with the error term $u$ since $X_2$ has been included in the specification, we end up with

$$Y=\beta_0 + \beta_1Z + \beta_2X_2 + u$$

for which the OLS estimator will be asymptotically consistent, since (writing $\mathbf W = [1:Z:X_2]$)

$$ \hat \beta = \beta + \left (\mathbf W'\mathbf W\right)^{-1}\mathbf W' \mathbf u$$

and its consistency will depend on the following probability limit being zero

$${\rm plim}\mathbf W' \mathbf u ={\rm plim} \left[\begin{matrix} 
\frac 1n\sum u_i\\
\frac 1n\sum Z_iu_i\\
\frac 1n\sum X_{2i}u_i\\
\end{matrix}\right] \rightarrow  {\rm plim} \left[\begin{matrix} 
\frac 1n\sum E(u_i)\\
\frac 1n\sum E(Z_iu_i)\\
\frac 1n\sum E(X_{2i}u_i)\\
\end{matrix}\right]= \left[\begin{matrix} 
0\\
0\\
0\\
\end{matrix}\right]$$

holds because both $Z$ and $X_2$ are orthogonal/uncorrelated to the error term $u$.

In other words, the fact that the regressors $X_2$ and $Z$ are correlated with each other does not create any problem (as long as of course the co-linearity between them is not near-perfect), on the contrary it more generally justifies including both in the regression specification, making it "multiple regression" -it exists exactly for such cases.