Changing things like did you and bet you into sounds that one might attempt to represent in spelling as “didju” or “didja” and “betchu” or “betcha” is something that happens with all speakers of English. Because the reasons behind it are based on where in your mouth those sounds are formed and what happens when you move quickly from one to the next, similar effects can be found in many other languages besides English as well. It’s more about human physiology as it relates to the production of these sounds in the mouth than it is about English in particular.
This simple and common phonetic phenomenon goes by fancy terminology: palatalized affrication under the influence of yod coalescence. (Affrication is when a non-affricate sound is replaced by an affricate — that is, a stop plus a fricative — while yod coalescence is when a rising diphthong starting with /j/ causes the previous stop to become palatalized.) Wikipedia observes:
Yod-coalescence is a process that palatalizes the clusters [dj], [tj], [sj] and [zj] into [dʒ], [tʃ], [ʃ] and [ʒ] respectively (for the meanings of the symbols, see English phonology). The first two of these are examples of affrication.
Unlike yod-dropping, yod-coalescence frequently occurs with clusters which would be considered to span a syllable boundary. It thus commonly occurs before unstressed syllables. For example, in educate, the /dj/ cluster would not usually be subject to yod-dropping (in General American, say), as the /d/ is assigned to the previous syllable, but it commonly coalesces to [dʒ]. Below are a few examples of universal yod-coalescence.
- /tj/→/tʃ/ in most words ending -ture, such as nature /ˈneɪtʃər/
- /dj/→/dʒ/ in soldier /ˈsoʊldʒər/
- /sj/→/ʃ/ in words ending -ssure, such as pressure /ˈprɛʃər/ (also in words ending consonant+sure, consonant+sion, -tion)
- /zj/→/ʒ/ in words ending vowel+sure, such as measure /ˈmɛʒər/ (also vowel+sion)
English has two affricate phonemes, /t͡ʃ/ and /d͡ʒ/, but that’s not what these are here. They are not phonemic. Because this is a phonological effect, when it happens here it is not phonemic: just a phonetic allophone that doesn’t alter which word is perceived to have been said.
It’s notably common in the dialect we’ve come to call “Eschewary” English, where words like dune and Tuesday come out sounding like June and choose day — and where Estuary comes out sounding like Eschewary. :) Unless you drop the yod the way many North Americans do in those sounds, they’ll almost always eventually coalesce into an affricate like that.
This is a common process in many languages, not just English, because it happens as a result of the difficulty in clearly and cleaning moving from the stop to the yod (the /j/ if you would). Think of the words situation and soldier, where the palatalization has been going on for so long that it’s now completely standard and would sound funny if you didn’t do it.
To demonstrate how the same process that produces the palatalized affricate of English did you occurs elsewhere, let’s look at a bit of Romance phonology, where we find this same thing can also happen with such combinations as Spanish en yerba in rapid speech by many native speakers where that turns into something that an English speaker might hear as the "j" of judge, which is really /d͡ʒ/.
This sort of thing happened historically as the yod that appeared in Proto-Romance changed how things were said (and eventually, spelled) compared with classical Latin. Although English is not a Romance language, the many Latin words it adopted from Norman French had lots of these effects going on in them.
In Romance you can see these yod effects both historically and today. Historic examples include Portuguese chover < VL. plovere (think of the fancy word pluvious in English) and Spanish llover from the same origin. The modern-day effects can be seen in how at times the /j/ sound in Spanish, including in the word yo spoken emphatically and in llover as pronounced by most yeísta speakers, can easily enough become affricated into [ɟ͡ʝ], a sound that English speakers hear as the /d͡ʒ/ in judge. There are many old jokes about Spanish speakers pronouncing in Yale the way an English speaker would pronounce in jail because of this. This happens within Spanish words, too, like enyesar which phonemically matches its spelling /enyeˈsaɾ/ but phonologically can come out as [e̞ɲɟ͡ʝe̞ˈsaɾ]. Brazilian Portuguese is famous for palatalizing the ends of words like frente and tranquilidade, which is much the same sort of thing as happens in English bet you and did you.
So this palatalization process of yod coalescence that produces fused affricates occurs naturally not just in English but in many languages. It is both an historical process and an ongoing one that continues to this day.