I think this is a good sentence:
(1) “The right side of the plane was badly burned in a fire after the plane careened through a field [...] like a four-wheeling Jeep.”
To hear a prescriptivist tell it, though, this is an awful sentence, symptomatic of an American carelessness for the meaning of words. In fact, let’s let a prescriptivist tell it. I quote here from James Cochrane’s vituperative volume Between You and I, page 23:
“Users of this expression may be surprised to discover that the original meaning of to careen is to turn a vessel over on its side [...] What they presumably mean to convey is the idea of something rushing headlong down a street on a dangerously erratic course. [...] In American English, careen is certainly a Lost Cause, since its use in this erroneous sense is recognised in dictionaries, but for British English it may not be too late to rescue it.”
Cochrane is right, though unbearably pretentious, in his claim that the first meaning of careen was to turn a ship onto its side, usually in order to clean off the barnacles and other sea-junk. This sense is attested all the way back to 1600 in the Oxford English Dictionary, and it continues to be used to the present day, albeit sparingly and primarily by seafarers.
But Cochrane stops short of the whole story. Careen has a second meaning that is just as uncontroversial as its original meaning: to lean, heel over, list, or tilt. Thus we get the following usages:
(2a) In Worcester a charming reminiscence hangs about the sloping heights of Asnebumskit, whose great hill-sides, which the Senator has bequeathed in trust to his two grandchildren, careen toward the city. 
(2b) The ship staggered, careened, and reeled, as wave after wave came thundering on her. [1863, cited in the OED]
It’s still occasionally used with this meaning today, at least in nautically-themed historical romance novels. But Cochrane makes no mention of it, and therefore, to use a nautical idiom, he’s selling us a bill of goods. He’s smugly saying that we are — as he puts it in the Preface to the 2nd Edition of the book — “half-educatedly” using careen, and yet he is only half-educatedly explaining the history of careen. You can almost see why he’d be irritated by people taking such a huge semantic leap with a word, moving straight from “turning on a side to clean” to “rushing unsteadily”, if that were the case. But it wasn’t. There were two reasonable semantic jumps, first from “turning on one side” to “tilting from side to side” (2b), and only from there to “moving quickly and unsteadily/uncontrolledly”.
But tons of words no longer mean what they once meant. (I wrote a few months ago about the curious history of awful and awesome, for instance.) So why does careen draw prescriptivists’ ire in a way that most other words whose meanings have changed do not? The answer is a single word: career. Prescriptivists are convinced that when we say careen, we really mean career. After all, the OED tells us that career means “to gallop, run, or move at full speed”. And that is very close to what we intend with careen. Very close indeed… and yet, not close enough.
The verb career apparently comes from a Latin root related to racing. The early uses of the word in the OED are about just this: the careerers are horses hurtling down the track fast as they can. Now, I am not an equestrian myself, but I generally think of horses as graceful creatures. For instance, I have watched two consecutive runnings of the Kentucky Derby, and the horses ran beautifully, steadily, and speedily both times. In neither case would I be willing to say that the horses careened down the track. On the other hand, I have also had a few opportunities to watch toddlers race. They most assuredly careen. They do not career. Perhaps this is strictly my own usage, but I think of careering as “moving swiftly and effectively”, and careening as “moving fast and uncontrolledly”. For me, the two have very closely related, but distinct, meanings. And this is why I continue to use careen.
I feel that this is the appropriate time to point out a line from Between You and I‘s Preface, where Cochrane is spelling out the purpose of his book:
“It is not written from any inclination to be purist in the sense of proposing that change is to be resisted at all costs; on the contrary, not only is change inevitable but the constant renewal and enlargement of our wonderful language is generally to be welcomed rather than not.”
You are welcome to draw your own conclusions about whether Cochrane had recently read his book when he wrote that preface.
Summary: Prescriptivists claim it’s wrong to use careening to mean “moving quickly and unsteadily”, because careening actually means “turning a boat on its side” and the intended meaning can be gotten from the obscure verb career. However, career does not mean the same as careen, and a series of simple semantic extensions can explain why careen has taken on its unsteady-motion meaning.