What you need to do is that if you move to the right(with your mouse) then also rotate the face of your players body to the right. What you have now is that if you hold W and look straight with your mouse, then the players body moves forward, but once you turn your camera 90 degrees to the right, you only rotate the camera but not the whole players body. If you would to rotate the body at the same direction where you rotate the camera, then the "front" of the body would always be where you are facing with your camera.
If you look at the default first-person-controller that unity provides, then go inside the mouse-look script and find the bit that tells you that it rotates the body every time you move the camera. They //comment every section of the script.
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