electromagnetism - How can relativistic effects be the cause of magnetic forces when the normal speeds of carrier charges are so low?

I had read here that magnetism arises from a current because of the special relativistic effect associated with the speed of the moving charges in that current. However that speed is only on the order of 0.024 cm/s, essentially zero compared to the speed of light, so there cannot be a relativistic effect to consider. Is special relativity the origin of magnetism?