For example, Thirteen (and not Ten-three), compared to Twenty-three, in English, Dreizehn, compared to Dreiundzwanzig, in German, Treize compared to vingt trois, in French, even dekatreis compared to eikosi tria, for greek.
This isn't found in most other areas of the world: In mandarin, 13 is essentially "ten three" and 23 is "twenty three", and Japanese, sanskrit, Thai, Swahili, Somali, Sotho, among others, follow this sort of method, to some degree
There are actually 2 main structures for 11-20 numbers in IE languages in Europe: the Germanic/Latinate pattern we see in thirteen, dreizehn, treize, and a 'Slavic' pattern as in Russian trinadcat' 'three-on-ten', where the 'ten' element is usually reduced (cf Russian desjat' 'ten' but Albanian--IE but not Slavic--dhjetë~trembëdhjetë). It might look like a minor distinction, but it's meaningful.
I don't think there's a satisfactory answer as to why we see the 'three-ten' pattern in some places, the 'three-on-ten' pattern in others, and 'ten three' in others: people started counting in a particular way, those patterns became entrenched among their linguistic descendants, and the forms became reduced as the patterns were entrenched. That said, these things do change--Romanian, for instance, uses the 'Slavic' pattern, treisprezece. But ultimately it's not a very significant difference.
As a French specific follow up: why does French not have a seventy (soixante dix - literally sixty ten) eighty (quatre vingt - literally four twenty) or ninety (quatre vingt dix - literally four twenty ten which leads to my all time favourite number: quatre vingt dix neuf -- four twenty ten nine = ninety nine)
This doesn't specifically answer your question about European languages in particular, but from a linguistics perspective, words that are used frequently tend to resist regularization. So for example, less common verbs in English follow simple "-ed" conjugation in the past tense: stay/stayed, cook/cooked, burn/burned but common words tend to follow archaic patterns or are based on strange happenstance: go/went, come/came, eat/ate, see/saw. This is (almost definitely) because people simply say these words more often and are more resistant to changing what they say all the time.
In daily life, we count to low numbers much more often than we count to high numbers, so this could explain why lower numbers resist regularization.
But that doesn't explain why it happens in some languages and not others. I won't try to give specific reasons, because I don't know, but one possibility is direct language reform by a government agency.
Slight correction, in greek only 11 (enteka) and 12 (dodeka) are irregular. 'Deka-tria' (13) and on is a regular pattern.