Let $G$ be a Lie group acting smoothly and freely on a smooth manifold $M$. Suppose that the quotient space $M/G$ is a topological manifold. Do we have $$\dim(M/G)=\dim M-\dim G?$$

Notes: This question was posted here on MSE. If $G$ acts properly, then of course the answer is yes. More generally, Daniel Robert-Nicoud observed here that the answer is also yes if there is a smooth structure on $M/G$ for which the projection $M\to M/G$ is a smooth submersion.