Can you give some examples of counter-intuitive phenomena discovered by linguists?

Almost everything a linguist will know about language will be counterintuitive to many native speakers. I've spent years teaching grammar to native speakers of English as well as intro to linguistics so I have come across a lot of things that completely dumbfound people.

Here are some areas to look into:

Phonetics and phonology

• Schwa is the most common sound in English!
• English has 18 vowels (vs. 5 vowel letters)
• The very concept of a phoneme seems to stump people

Lexicon and Etymology

• The extent to which words have changed
• The enormous polysemy of many words
• The arbitrariness of the sign vs. motivation

Syntax and semantics

• How quickly language becomes intractable when treated algorithmically (something the whole field had to learn)
• How idioms work

Pragmatics

• Speech acts are normal and not a misuse of language
• Presupposition is part of meaning

Sociolinguistics

• There's no clear boundary between dialect and language
• There's no such thing as a 'primitive language'
• Standard English is just one dialect of English
• Bilingualism is the norm not the exception
• Children are not harmed by learning multiple languages at once
• Code switching is normal

I've written a post about 5 things people should know about language and another one with 17 things linguists know about language. Most of those will come as a surprise to non-linguists (and even some linguists raised in certain traditions).