Your average person here needs a car to get to work, get groceries, drop kids off to school and to do anything else that's importance. You cannot design cities such that you need a car to get around, and then go and punish people for using a car, it makes zero fucking sense.
In theory yes, the point of a carbon tax is to internalize an externality, but most people will just swallow the cost anyways because the train systems are shit, the buses are shit, and you designed cities so that you need to drive 15 minutes to get to a grocery store.
If anything, all a carbon tax will do is reduce consumer spending and damage the GDP without affecting at all how people get around.

If the government really wanted people to stop using cars, they should instead focus on better public transport instead of punishing people who have no other choice.