Is something wrong with the British Broadcasting Corporation now?
Why is ‘long’ a short word?
Why is ‘abbreviation’ a long word?
Because those words are heterological.
A quality response. 
As we’ve slightly side-tracked, here’s some more use of language:
A contradictory oxymoron is a tautology. 
Yeah! Speaking of odd spellings, they constantly transpose or add extra, unnecessary letters: theatre, colour, aluminium, cheque, etc. And don’t get me started on the incorrect word usage! Cookies are biscuits, fries are chips, chips are crisps. It’s madness, I tell you! 
I say!
We Brits spoke English first - so you Americans changed the spelling - not us!
Bad teachers, bad students.
Just sayin’
Yeah, we fixed English!
Kidding-- as the son of a mother who emigrated here with my grandparents from London, I grew up with British English, and I’m well aware we Americans dumbed English down.
To be fair, I sometimes wish our English spellings were like the American simplified versions…
Phonetic alphabet?
The English, of course, changed the spelling before the Americans did. By intention, to change away from phonetic, towards etymology (sometimes with no valid rime or reason)
I’m going to steal this.
I offer it to you freely! 
One I hate is on crime/medical shows they often refer to GSWs - which has more syllables than gunshot wound.
Can imagine writing GSW, but not speaking it.
I asked that same exact question.
I was going to say last year but I see it’s been since '21. Time flies…
I believe, rather than “wrong”, you’ll find them rather satisfied.
Rarely? Oh no. All too frequent a spelling.
Maybe in parts of the south; but it’s overwhelmingly spelled with a “C” in most of the US. And often when you see it spelled with a “Q” it’s for marketing reasons.
It’s rare enough in my experience that “barbeque” just looks extremely weird and wrong. It’s also not considered proper in any formal communication. It’s the written equivalent to the word “ain’t”; it exists and in some areas it might even be common, but it’s always “wrong” (from a prescriptivist perspective at least).
If you Google both spellings, which one gets more hits?
Here’s a Google Ngram of both spellings:
In case that’s hard to read, the blue line is “barbecue” while the red line is “barbeque”.
I’ve seen it fairly often, but it looks weird to me too. I can’t decide whether to pronounce it “bar-beck” or “bar-be-quee.”