In special relativity length contraction occurs in the direction of travel for an object.
However, since physics is a science of observations, has it ever been observed not to simultaneously occur in the object's other two dimensions or is it just always assumed not to occur?
If it doesn't occur, why not?
Imagine two vertical toothpicks, each with a little light at its top and bottom, passing each other in the horizontal direction. Each toothpick can tell whether its top and bottom lights pass over or under the other toothpick, or whether they hit the other toothpick and are reflected back.
Therefore the toothpicks are able to agree on which one is shorter.
But if there were a relativistic length contraction, each would say the other is shorter than itself, contradicting that agreement.
However, since physics is a science of observations, has it ever been observed not to simultaneously occur in the object's other two dimensions or is it just always assumed not to occur?
The Michelson Morley experiment (MMX) was originally explained as due to length contraction by Lorentz. This was prior to Einstein’s development of special relativity, but the experiment can still be understood as evidence for length contraction. However, the MMX is not compatible with length contraction in all three directions.
Another thought experiment that might help you to see the contradiction if objects contracted along directions perpendicular to their motion.
Imagine 2 identical hollow cylinders moving towards each other along their axis. Now, if objects contracted along the perpendicular directions,
then according to the frame of cylinder A, cylinder B will have a smaller radius and hence cylinder B will pass inside cylinder A.
But according to the frame of cylinder B, cylinder A will have a smaller radius and hence cylinder A will pass inside cylinder B.
But, these two cannot both be true. You cannot have both A passing through B and B passing through A. This contradiction hopefully shows you why it is not possible that objects contract along the directions perpendicular to their direction of motion
