Subgroup generated by conjugacy class normal?

Is a subgroup generated by a conjugacy class normal?

I have been trying to look for a counter-example. Naturally, I thought that simple groups would be suitable. However, the generated subgroup seems to always be the whole group itself, hence trivially normal. I have tried this with $A_5$.

Also, this tells us that the generated subgroup would be the normal closure, which intuitively sounds too strong. I have tried proof by contradiction, both from the positive and negative position, but it leads me nowhere.

Any hints and directions are appreciated.

$A_5$ is simple so you will not find any interesting normal subgroups. Let me give you a hint: is $g,a,b \in G$, then $g^{-1}abg=g^{-1}ag \cdot g^{-1}bg$. This should help you proving that the subgroup generated by a conjugacy class is actually normal.
Hint: Let $N$ be a subgroup generated by $<g^x|x\in G>$. Take a $n\in N$.
Then $n=h_1h_2...h_k$ for $h_i=g^x$ for some $x\in G$. Now what can you say about $n^y$ for some $y\in G$ ?