What is the reason to believe that the laws of physics are same in all frames of reference?

Is there any supportive evidence which suggests so other than the evidence of common sense and intuition?

Yes, there is quite a substantial body of experimental evidence supporting the two postulates of relativity. The best summary I know is this page:

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

It is important to understand how these experiments are analyzed. Since you would like to test relativity you cannot assume it or its postulates. Instead, you create a test theory that is more general. It does not assume either of the postulates, but has one or more free parameters that can be adjusted. These free parameters have some specific setting which produces relativity, and any other combination of free parameters indicates a violation of relativity.

Then you use this test theory to develop experiments that measure the value of the free parameters. If the measured result of the experiment is equal to the relativity value (to within experimental error) then the experiment validates relativity. This approach is important because it allows you to test a theory without assuming it.

For special relativity Zhang (Special Relativity and Its Experimental Foundation https://doi.org/10.1142/3180 ) describes a general test theory, and for the standard model the Standard Model Extensions is a complete test theory including all possible relativity violating terms.

If special relativity were not true, the everyday world we experience would be very different, in very obvious ways. I will provide just one simple example:

Special relativity is embedded in Maxwell's equations and tells us (among other things) that if we insert a magnet into a coil of wire, so as to induce a current to flow in it, there will be no way to tell from looking at the meter measuring that current whether it was the coil that moved or the magnet that moved.

If this was not the case, then the output of an electric generator would depend on whether the armature inside it were rotating and the case was fixed in position, or the armature was fixed and the case were rotating. No such effect exists.

In addition, the output of an electric generator would depend on its state of motion relative to some fixed reference frame attached to the universe as a whole. As the generator, which is fixed to the surface of the (rotating) earth, moved relative to that frame, its output would vary according to its angle relative to that frame. This effect does not exist either.

The laws of physics being the same in all inertial reference frames is an idea that originates not with Einstein but Galileo, who noted if you're in a sealed windowless room on a ship you can't tell whether the ship is moving (though you can tell if it's accelerating, such as when it bobs). Special relativity differs from Galilean relativity only in how we convert between inertial reference frames, claiming the transformation of spacetime coordinates to be Lorentzian instead of Galilean. With the right assumptions, you can show only that one of these transformations is correct. The Galilean case is then the special case $c^{-2}=0$, which is measurably incorrect. The positive empirical value of $c^{-2}$ is known precisely enough to define the metre in terms of the second.