Here is a counterexample: Let $a,b$ be distinct units in $k$ such that $ab \neq 1$, and let $G = SL_2 \times SL_2$ be given the usual block diagonal embedding into $GL_4$. Then the matrices
$$A = \left( \begin{smallmatrix} a \\ & 1/a \\ & & b \\ & & & 1/b \end{smallmatrix} \right) \qquad \text{and} \qquad B = \left( \begin{smallmatrix} b \\ & 1/b \\ & & a \\ & & & 1/a \end{smallmatrix} \right)$$
are conjugate in $GL_4(O)$, but not $G(O)$. The underlying problem in this example seems to be that the normalizer of the torus in $G$ is too small.

I suspect Mark Palm's suggestion for $G = SL_2 \hookrightarrow GL_2$ will admit a counterexample when $k$ has characteristic 2, but I have not worked it out.