Not if you drive it very much/for very long. See this graph [1] (from this article [2]) for instance. Note that they're evenly diving 173,151 miles across the 13 "years" (and don't ask me why they decided to make the x-axis "years").
And that's with a modern fuel efficient car, not some ancient one.
[1] https://graphics.reuters.com/ELECTRIC-VEHICLES/EMISSIONS/rlg...
[2] https://www.reuters.com/business/autos-transportation/when-d...
Driving emissions numbers I remember off the top of my head are Swedish averages, around 2.5 tonnes CO2e per year (15000 km/year). This is averages for the Swedish car fleet, which tend to be smaller models and more modern than many countries.
So: sure, production emissions are a big factor, but driving the car can easily win the efficiency savings back in a fraction of the car’s lifetime.
On the other hand I also know people who drive a multiple of that.