OLS
The OLS part is clear: $\sum_{i=1}^n y_{2i} (y_{1i} - \hat\gamma_{12} y_{2i} - \hat\beta_{11} x_{1i}) = 0$ and $\sum_{i=1}^n x_{1i} (y_{1i} - \hat\gamma_{12} y_{2i} - \hat\beta_{11} x_{1i}) = 0$, where $\hat\gamma_{12}$ and $\hat\beta_{11}$ are the OLS estimators.
2SLS
For 2SLS, I will assume that $(x_{1i}, x_{2i}, x_{3i})$ is used as IV. The corresponding 2SLS is identical to the IV estimator using $\hat{y}_{2i}$ and $x_{1i}$ as IV. Thus, (I think) the "normal equations" are
\begin{align}
\sum_{i=1}^n \hat{y}_{2i} (y_{1i} - \tilde\gamma_{12} y_{2i} - \tilde\beta_{11} x_{1i}) &= 0,\\
\sum_{i=1}^n x_{1i} (y_{1i} - \tilde\gamma_{12} y_{2i} - \tilde\beta_{11} x_{1i}) &= 0,
\end{align}
where $\tilde\gamma_{12}$ and $\tilde\beta_{11}$ are the 2SLS estimators using $x_{1i}, x_{2i}, x_{3i}$ as IV, and $\hat{y}_{2i}$ is the fitted value obtained by regressing $y_{2i}$ on the instruments $x_{1i}$, $x_{2i}$ and $x_{3i}$.
In the above, I understood 2SLS as the IV estimator using $(\hat{y}_{2i}, x_{1i})$ as IV. We can also understand 2SLS as the OLS estimator of $y_{1i}$ on $(\hat{y}_{2i}, x_{1i})$. In that case, the normal equations can be written as
\begin{align}
\sum_{i=1}^n \hat{y}_{2i} (y_{1i} - \tilde\gamma_{12} \hat{y}_{2i} - \tilde\beta_{11} x_{1i}) &= 0,\\
\sum_{i=1}^n x_{1i} (y_{1i} - \tilde\gamma_{12} \hat{y}_{2i} - \tilde\beta_{11} x_{1i}) &= 0,
\end{align}
where $y_{2i}$ is replaced with $\hat{y}_{2i}$. The two sets of normal equations for 2SLS (one in terms of $y_{2i}$ and the other in terms of $\hat{y}_{2i}$) are identical.
Intercept
I have noticed that the intercept is excluded in the question. If the model is $y_{1i} = \beta_{10} + \gamma_{12} y_{2i} + \beta_{11} x_{1i} + \epsilon_i$ instead, you need equations for the intercept as well.
Matrix notations
Using matrix notations, let $y$ be the $n\times 1$ matrix of the LHS variable, $X$ the $n\times 2$ matrix of $(y_{2i}, x_{1i})$ and $Z$ the $n\times 3$ matrix of $(x_{i1}, x_{i2}, x_{i3})$. Let $\beta = (\gamma_{12}, \beta_{11})'$. (I am considering the model without the intercept. For the model with the intercept, $X$ and $Z$ contain a column of ones.) Then the OLS estimator is $\hat\beta = (X'X)^{-1} X'y$, which solves the normal equations $X'(y-X\hat\beta)=\boldsymbol 0$. To see this, note that $$X'(y-X\hat\beta)=\boldsymbol 0 \\
X'y - X'X\hat\beta = \boldsymbol 0\\
X'X\hat\beta = X'y\\
\hat\beta = (X'X)^{-1}X'y$$
The 2SLS (using $Z$ as instruments) estimator is $$\tilde\beta = [X'Z(Z'Z)^{-1} Z'X]^{-1} X'Z(Z'Z)^{-1} Z'y = (\hat{X}'X)^{-1} \hat{X}'y$$ where $\hat{X} = Z(Z'Z)^{-1} Z'X$. This 2SLS solves $\hat{X}' (y-X\tilde\beta)=\boldsymbol 0$, the normal equations for this 2SLS. Again, to see this, note that
$$\hat{X}' (y-X\tilde\beta)=\boldsymbol 0\\
\hat X'y - \hat X'X\tilde\beta = \boldsymbol 0\\
\hat X'X\tilde\beta = \hat X' y\\
\tilde\beta = (\hat X'X)^{-1} \hat X'y $$
You will see that $\hat{X}$ is the $n\times 2$ matrix of $(\hat{y}_{2i}, x_{1i})$. It is also true that $\tilde\beta = (\hat{X}'\hat{X})^{-1} \hat{X}'y$, and the normal equations are also written as $\hat{X}' (y-\hat{X}\tilde\beta)=0$. The two different expressions are identical because $\hat{X}'X = \hat{X}'\hat{X}$.
