No, toothpaste will not remove car scratches.
While toothpaste is a mildly abrasive material, it is not abrasive enough to remove scratches from a car's paint job. In fact, using toothpaste on a car scratch can actually make the scratch worse.
If you're looking to remove scratches from your car, you're better off using a dedicated scratch remover product. These products usually contain abrasives that are much finer than the abrasives found in toothpaste, meaning they'll be less likely to damage your car's paint job.