So how can we say or prove time dilation in a moving frame or disprove simultaneity using this thought experiment? Am I wrong somewhere?
Let’s step away from thought experiments for a moment and look directly at the Lorentz transform. In the usual configuration the x direction is the direction of relative movement and the x’ axis is parallel to the x axis and similarly for the other axes. So the Lorentz transform is:
$$\Delta t’ = \gamma \left(\Delta t -\frac{v \Delta x}{c^2}\right)$$ $$\Delta x’ = \gamma ( \Delta x - v \Delta t)$$ $$\Delta y’ = \Delta y$$ $$\Delta z’ = \Delta z$$
In your scenario x is the direction along the road and y is the direction across the road. Since in your scenario $\Delta t=0$ and $\Delta x=0$ we can plug into the first formula and find $\Delta t’=0$.
So it is expected that both frames agree that they happened simultaneously. This thought experiment is not capable of studying the relativity of simultaneity.
Similarly, for time dilation since $\Delta t’$, $\Delta t$, and $\Delta x$ are all zero then the first equation reduces to $0=\gamma 0$ which is true regardless of $\gamma$ and therefore provides no information about time dilation.
This scenario is set up in a way that it is not sensitive to either time dilation or the relativity of simultaneity. $\Delta x=0$ makes the relativity of simultaneity irrelevant and $\Delta t =0$ makes time dilation irrelevant