I think the three-and-ten, four-and-ten way of expressing numbers is primarily an ancient Greek thing. The modern numbers are expressed differently (δεκατέσσερα / dekatessara for fourteen, for example). In a lot of older European languages 11 and 12 behaved irregularly. You could argue that they do in English too (we don't have oneteen and twoteen).
I haven't read of any particular reason for this, but I'd posit that numbers up to twelve were more commonly used in everyday life, so shorter, irregular forms were easier to use and remember. Much like many of the irregular verb forms in spoken language happened because they were so commonly used.
The ancient Greek system also gave us triskaidekaphobia - the fear of the number 13.
French has some weirdness to its counting. 11 - 16 is specific words, 17 - 19 is "ten {number}" (dix-sept, dix-huit, dix-neuf).
There's also some fun when you get to 70... which is soixante-dix (sixty ten)... and 80 which is quatre-vingts (four twenties)... and 90 as quatre-vingt-dix (four twenties ten).
Some dialects of French outside of France have changed how they count.
>we don't have oneteen and twoteen
But we do have duodecim, aka dozen. So we just need to add "onezen" to the current vernacular to make the set complete. Let's go team onezen!
There were plenty of base-12 number systems in Europe, one way of counting to 12 is using your thumb to count the bones in the other 4 fingers, one hand for the 1's digit, the other for the 10's digit.
12 being divisible by 2,3,4,6; 10 being divisible by 2,5 -- means base 12 is easier to multiply and divide to reach whole numbers.
Indeed, in ancient greek they put the "δέκα" (ten) part second. Τρία και δέκα (13), τέσσερα και δέκα (14), πέντε και δέκα (15), etc, but 11 and 12 was (and still is in modern greek) irregular, έντεκα (enteka) and δώδεκα (dodeka) respectively.