I'm afraid that I don't think that it's possible for you to achieve what you want here.
Here are your criteria again:
How can I tell this coworker that I don't feel safe in their car when they drive like they normally do, and to drive safer when I'm in their car, without their feeling offended and my not losing my ability to carpool with them?
What you want is known as to "have your cake and eat it too".
You've worked there for a week. Your boss is doing you a favour by driving you to and from work. (Yes, it's your boss: a co-worker who's superior to you and to whom you report directly is also known as your boss.)
You want to tell your new boss in your new job (who's doing you a favour) that they're a bad driver and make you feel unsafe. But you don't want them to feel offended by this, neither do you want to have to find another way to get to work and back every day. I don't really see that happening.
However you word it, sugar-coat it, or use "I statements"; what they're going to hear is "You're a bad driver". No one likes to hear that. It's basically telling them that they're bad at what they're doing. That sort of thing makes people feel offended.
I think you have three options open to you:
- You can tell your boss as politely as you can that they're a bad driver, by following one of the other answers. What will happen next depends on your boss's character. It'll range from nothing changing, to your career opportunities being severely limited, and might involve you having to take cabs to work for a while until you get your own car.
- You can get your own car as soon as humanly possible. Solve the problem by removing the whole situation.
- You could try and work up to it over time. One week (one working week at that, so only five days) is not really long enough, in my opinion, to tell your new boss that they're a bad driver. If you build rapport with them, get more friendly with them over time, then you should arrive to a point where you can speak to them about this.
Personally I would go for the second option and get out of the situation where my life is in the hands of a dangerous driver.
Anecdote time! You have to be careful even with friends: I have a friend who's a bit of a crazy driver; not a very bad driver (not as bad as the driver in the question!), but not a very safe one either.
One day I installed an application on my phone, I can't remember the name, "onroad" or something like that (it was many years ago and there a lots of apps that do the same thing these days) that I'd read about somewhere, which would use your phone's camera to watch the road and objects in it (e.g. cars) while you were driving, and it would warn you if you were approaching any object too quickly.
I tried this app out, and I just couldn't get it to work: I'd drive a little bit less safely than I was comfortable with and slow down or brake at (what I considered) the last possible safe moment, but the app just wouldn't trigger.
The following week I was in my friend's car as passenger, and thought I'd try this app out again. It worked just fine - alarms went off all the time as she would brake to stop in a way that would make Elon Musk and his suicide-burning rockets proud.
I had the temerity to joke about this later on that day. That the app I thought was broken actually worked just fine, I just needed my friend to be driving. She was in a mood with me about it for the next two weeks! I think it was even longer before she drove me anywhere again. And this is a close friend.
The funny thing is that during our drive the app actually worked - in a passive-aggressive way - because the beeping noise from my phone wasn't me saying "you're a bad driver", it's an piece of software saying it, and everyone knows that pieces of software are hateful things. I could definitely feel her begin to drive less quickly and being to slow down sooner, to avoid having the application admonish her with a beep.