GoKawiilSiri's ability to speak notifications is great in the right circumstances, but if it won't shut up about incoming alerts in the car, you can turn this off. Back in Settings on your phone, choose Notifications > Announce Notifications. If you want to use this feature outside of CarPlay, keep the top toggle on and select the CarPlay menu. There, disable Announce Messages and Siri won't read anything that comes in during a trip.
Back on the Announce Notifications page, you might also want to turn on Reply Without Confirmation. By default, when you speak a reply, the assistant will read what you said back to you for confirmation before sending. While this helps avoid misspoken messages, it also creates extra steps behind the wheel. Turn this toggle on and sending a reply becomes a single step.
Another way to prevent stoppages with Siri is hidden under Settings > Accessibility > Siri > Require "Siri" for Interruptions on your phone. Normally, Siri will stop what it's saying when it detects you talking back. If your phone is doing this in error because of road noise or other people in the car, enable this option. Then, Siri won't stop what it's saying unless you say "Siri" to prompt it again. Similarly, if it tends to stop listening because you take too long to think, change Siri Pause Time to Longer or Longest to give yourself more time.
Although Siri is required for CarPlay, you can still turn off some forms of its activation. Go to Settings > Apple Intelligence & Siri > Talk to Siri on your phone and choose Off in the top menu. After doing this, you'll only be able to summon Siri by hitting the voice command button on your steering wheel or holding the Side button on your phone. And in some cars, using the volume knob (or your phone's volume buttons) when it's talking will adjust the volume for Siri only, while not affecting music or other playback. This depends on your vehicle, but is another trick worth a try if you want more control.