This road has been traveled before. What you say only makes sense if the new car has similar gas mileage to the older one.
From:
http://web.mit.edu/sloan-auto-lab/research/beforeh2/files/weiss_otr2020.pdf
From Table 5.4: About 75% of all emissions over the life of a conventional gasoline-powered 1996 car (Toyota Camry) are generated in use. Only about 7% of total emissions are generated during manufacture. What you say only makes sense if the new car has similar gas mileage to the older one.
The effect of better gas mileage over the life of a car is huge in terms of cutting total emissions:
1996 Camry sedan, manual transmission EPA MPG: is about about 23 mpg
source: Toyota
Conventional gasoline engine, mid-size or compact cars, best mileage is about 36 mpg
source: https://www.cars.com/articles/best-and-worst-gas-mileage-2016-1420683223195/
If same manufacturing methods and fuel delivery methods are observed, then the total emissions (including manufacturing, fuel production and delivery, emissions on the road) of the car over its life cycle is:
1996 Camry: grams Carbon/km = 71.8 g C/km; at 20,000 km/yr for 10 years: 14,000 kg
2016 fuel efficient compact or mid-size car: 51 g C/km; at 20,000 km/yr for 10 yrs: 10,496 kg
So, getting rid of an old car -- that is not a bad choice in terms of carbon emissions.