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Shimmy in steering

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Easiest way would be to throw a pair of good known wheels n see if goes away. Bad tires/wheels can cause weird issues...
Can always put car on the lift n shake things around n see if anything is off.

Also even normal tires can cause strange steering sense. I put winter tires on 20s n steering felt really loose/weird from center to 90deg both directions to the point of its hard to drive in the rain. With regular tires on 22s it feels really solid.