Show the category of Hausdorff spaces has Pushouts and Coequalizers

Question

Let $\mathbf{Haus}$ be the category of Hausdorff spaces. I'm required to show that $\mathbf{Haus}$ has Pushouts and Coequalizers.

I don't know if I'm even close, but here is what I've tried:

1. I tried with the regular construction of coequalizer, that is, for $f, g:X\to Y$, we build $Y/R$ with $R$ the minimum equivalence relation containing the related pairs $f(x) \sim g(x)$, for all $x \in X$. But this quotient is not Hausdorff space.

2. For $X \in \mathbf{Haus}$, we build $X/R$, with $$R = \{(a, b) \in X \times X: \forall Y \in \mathbf{Haus}, \forall f: X \to Y, ~~ f(a) = f(b)\}$$ Then, $X/R \in \mathbf{Haus}$, but it is not coequalizer.

3. Then, I found that if a category $C$ has initial objects and Pushouts, then there is a given way to build coequalizers. So I decided (to try) to build Pushouts. I have $f: X \to Y$ and $g: X \to Z$ s.t. $X, Y, Z \in \mathbf{Haus}$. Then I build $Y \amalg Z$ and then $(Y \amalg Z)/R$ with $R$ the generated equivalence relation containing $f(x) \sim g(x)$, for all $x \in X$. But, I cannot prove that $(Y \amalg Z)/R \in \mathbf{Haus}$.

4. I also tried two find a way to show that $(Y \amalg Z)/R \in \mathbf{Haus}$ by showing that $R$ is closed of $(Y \amalg Z) \times (Y \amalg Z)$. But I found no way to express $R$ as a finite union of closed sets (including the Diagonal).

Note. I noticed that if $(Y \amalg Z)/R \in \mathbf{Haus}$, then it is pushout and further more, it is coequalizer. But I cannot even prove that, indeed, $(Y \amalg Z)/R$ is Hausdorff space.

I also read this wikipedia construction (but it didn't clarify anything to me):

https://topospaces.subwiki.org/wiki/Hausdorffization

Any help, suggest or hint is really appreciated.

Additional posts:

Question about quotient of a compact Hausdorff space