discrete normal subgroup of a connected group
Suppose $h \in H$, $g\in G $ and $ghg^{-1}\not=h$, then since $G$ is connected manifold, hence path connected, one can find a path $g(t)$ in $G$ going from $e$ to $g$. Notice $a(t):=g(t)hg(t)^{-1}$ realizes a path lying entirely in $H$ from $h$ to $ghg^{-1}$ which contracts the discreteness of $H$.
Hint, See: Lecture V - Topological Groups, Theorem 5.5.