Is a codimension one closed subgroup of a connected linear algebraic group is automatically normal?

Question

Let $G$ be a connected linear algebraic group and let $H$ be a closed subgroup of $G$ of codimension one. Is $H$ necessarily normal?

Answer 1

No. Try taking a look at the $ax+b$ group -- a.k.a. the group of one-dimensional affine linear transformations -- over your favorite field, for instance.