Let $H$ be a closed, connected subgroup of an affine algebraic group $G$.
Then is it true that for every $x\in G$, the quotient spaces $G/H$ and $G/xHx^{-1}$ are isomorphic as varieties ?
Where I am considering $G/H$ as a variety in the usual sense as described in Humphreys, Linear Algebraic Groups, Chapter IV
By the universal property of quotients of algebraic groups, the composition $G \xrightarrow{\operatorname{Int} x} G \rightarrow G/xHx^{-1}$ induces a morphism of varieties $G/H \rightarrow G/xHx^{-1}$. Similarly you can obtain a morphism $G/xHx^{-1} \rightarrow G/H$ and check that these are inverse to each other.