I have an argument, which I wonder if someone could check:

Let $X$ be a irreducible reduced scheme over a field $k$. Then we have a normal scheme $X^{norm}$ with a finite birational $f:X^{norm}\rightarrow X$. Then for any étale sheaf $\mathcal{F}$, the higher direct image $R^if_*\mathcal{F}$ vanishes for $i>0$. In particular, the Lerray spectral sequence yields $$H^p_{ét}(X,f_*\mathcal{F})=H^p_{ét}(X^{norm},\mathcal{F}).$$ However, I would like understand $H^i_{ét}(X,\mathbb{Z}/n)$ and relate it to the cohomology of $X^{norm}$. However, since $X$ is assumed to be irreducible, and $f$ is assumed to be surjective, we know that the pushforward of $\mathbb{Z}/n$ is constant (see https://math.stackexchange.com/q/1352862). Hence we have $$H^p_{ét}(X,\mathbb{Z}/n)=H^p_{ét}(X^{norm},\mathbb{Z}/n).$$ In particular, passing to the limit, we get that the $\ell$-adic cohomology is invariant under normalization.