Equivalence between fibre bundles

Question

Let $F$ and $U$ be two topological spaces on which a topological group acts (by the left) and consider the product action $G\times(F\times U) \rightarrow F\times U$ defined by $g(f,u)= (gf,gu)$. Now every left action can be written as a right action by defining $xg:=g^{-1}x$. So my question is, does this imply that the bundle $(F\times U)/G$ over $U/G$ is isomorphic or equivalent to the twisted product $F\times_G U$?