What do you do if you live in a small city, but travel to a major city and have to drive?

If you can't handle traffic you simply don't go to a major city. Or you drive in and leave your car and jump on public transportation. 

You will get use to it,

My daughter didn’t drive until she was 20. She moved to a different state after she graduated college and had to drive in major highways for the first time. She had a 45 minute drive into work twice a week that was nerve racking for awhile

Do the drive on a Sunday at mid morning with lower traffic. Practice the drive so you know the road and where you are going.  That can make you more comfortable when there is heavier traffic. Do the drive several times before you need to drive into work. 

Driving on city streets is really about the same in a small place or a big one. Yes, there are more cars to watch for, but the roads and the signals are essentially the same. Multi-lane highways are only more difficult when it comes to getting on and off, and a few simple things will minimize those difficulties. Don't change lanes any more than you have to. Only do so to get into the lane you want, which is the right or centre or centre/right lane, depending how far you're going.

You can find tutorials and advice on driving mutli-lane highways online. Maybe take someone with you who can help navigate. Arrive and depart when it's not rush hour. Plan a route that gets you off the 12-lane as soon as possible.

Try traveling to a bigger but not a major city first.  The main thing in big cities is being alert and knowing where you want to turn.  

Then don't drive.

Take a Taxi or Uber/Lyft or public transportation. 

I have been in all the big cities in the USA and live in the 2nd largest metro area in the world.  I have never seen 12-lane highways. 

There are a few options:

*Use public transportation so that you don't have to drive.  Most major cities have plenty of transportation options, and in some cases it's far easier and cheaper to use that instead of driving and paying for parking

*Visit at a time when things are a little slower, or at least time your driving through the city for less busy times of day

*Use this as an opportunity to learn how to drive in the big city

Test the route at weekends / quieter times to get used to the road layout.

Or get the train.