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In short, you lean inside to reduce horizontal acceleration so that your weight is contributing as much as possible to the normal force (perpendicular to the surface of the track). Since friction force is coefficient of friction * normal force, if you are perpendicular to the surface you maximize friction and therefore the amount of grip you have. And because you are turning inwards, you are being "pulled" outwards, which balances the horizontal force caused by leaning inwards.

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