How would time dilation and space contraction work? If you are at rest in your car and you emit a flash, you see it propagating out at $c$ in a spherical shell.
What if you are moving at $v=0.99c$? Well you already are in the reference of some cosmic ray. Did you notice the time dilation and space contraction? No.
You still see a shell expanding at $c$. Now an observer riding with the cosmic ray would also see a shell of light expanding spherically at $c$. The only way time dilation and space contraction enter the picture is if the cosmic ray riding observer used a Lorentz transformation to confirm that you also see a shell expanding at $c$ in your reference frame, but of course, your observations are completely independent of an imaginary (or real) observer looking at you from a boosted frame.
The reason you both see a spherical shell of light expanding at $c$ in each of your own reference frames (or any frame) is because the position of the light-surface is space-like separated from the observer, so that simultenaety is velocity dependent. "Where" the light is now depends on "when" is now, and that is boost (magnitude and direction) dependent in such a was that all observers see light propagating isotropically at the speed of light.