Is $L^2(\Omega) =L^\infty(\Omega)$ for bounded domains?

Is it true that a function $f$ on a bounded domain $\Omega\subset\mathbb{R}^n$ is $L^2$ if and only if it is $L^\infty$ on that domain?

Suppose $f\in L^2$

then $f^2$ is integrable, so it is finite on sets of positive measure. So its essential supremum is finite, right?

Suppose $f\in L^\infty$

so it has a finite essential supremum, so i can estimate it$$\text{ess sup} f \leq g \text{ a.e.}$$

for a constant function $g$ that i just define as the essential sup of $f$:$$g:= \text{ess sup} f$$

and $g$ is just a constant function of finite value over a bounded domain which is $L^2$, right?

I have a feeling that $L^2(\Omega) \neq L^\infty(\Omega)$, so where is my mistake?