Solow Model, Growth rate of K/L and Y/L in steady state
If $Y = C \cdot X$ where $C$ is constant and $\frac{\dot{X}}{X} = g$ then we can solve for $\frac{\dot{Y}}{Y}$ as follows:
$$ \frac{d}{dt} Y = \frac{d}{dt} C \cdot X = C \cdot \frac{dX}{dt} = C \cdot \dot{X} \Rightarrow$$
$$ \frac{\dot{Y}}{Y} = \frac{C \cdot \dot{X}}{ C \cdot X} = \frac{\dot{X}}{X} = g$$

Therefore, as you concluded, they both grow at rate $g$.