I wasn't gonna enter this fray, but I've decided to do some digging.

First of all, what's a contraction? There are plenty of definitions out there, and, taken as a whole, they leave much wiggle room.

I don't know how authoritative this is, but I liked the way contractions were described in a link found in an earlier answer:

A contraction is a word that is formed by combining two or more words which often occur together in speech. In the process of this combining, one or more segments (i.e. vowels and/or consonants) of the component words are phonetically altered, reduced, or omitted entirely.

It goes on to divide contractions into two families:

• Dictionary-recognized contractions, which are "well established and are included in dictionaries, such as do not → don't, I am → I'm, it is → it's, we would → we'd."

• Informal contractions, which are "not in most dictionaries," and "beyond the recognized contractions that are acceptable in writing ... such as going to → gonna, want to → wanna, should have → shoulda, have to → hafta, kind of → kinda, sort of → sorta.

Put that way, I have no problem labeling these as informal contractions. However, one ELUer went so far to say that this categorization was "horribly wrong ... and will therefore misinform thousands of readers in the future." (That sounds a bit melodramatic to me, but that's what the comment says.)

Up in the comments to the original question, the what's-a-contraction conflagration continued:

betcha is not at all a contraction for "bet you". The form has become kind of a snowclone (or whatever the ... best term is). We now GENERATE words "in this style", such as lotsa and so on. – Joe Blow

@Joe Blow Betcha it is. Sometimes, at least. – Euan M

So, I looked up the troublesome betcha on OneLook, and followed the links. A couple dictionaries indeed labeled this a contraction, while others carefully danced around the contraction label, using terms like informal, eye dialect, slang, and spoken. Although it doesn't get its own listing in the OED, it does get a mention there as a "corrupt" form of bet you.

For what it's worth, OED does list shoulda (as "repr. colloq. or vulgar pronunc. of should have") while hafta also gets its own entry (and is labeled with "Representing a colloquial or regional pronunciation of have to"). The OED lists coulda as a "colloq. shortening of (I) could have."

None of those OED entries mention the c-word (contraction); shoulda and hafta get dissected in the OED as one being vulgar, and one being regional. The overlaps between terms like "informal" and "vulgar" were discussed in this ELU question ("informal" > "colloquial" > "slang" > "vulgar"), but the term regional only gets mentioned in a couple comments there, one of which says:

I believe "colloquial" has connotations of "provincial" or "regional", i.e. informal language which is peculiar to a certain geographical area.

But the other reads:

A colloquialism is not regional but conversational, typically with an informal context.

Back to the original question:

In linguistics, is there a term describing this phenomenon, i.e., when the syllables of two words are slurred together in the spoken language? They are not contractions.

I don't think there is a term describing this, but I think there are several that can be used. Lemme list a dozen:

• informal contractions¹
• colloquialisms²
• eye dialect³ (when this slurred-together speech is converted to written form)
• a contracted form that is not acceptable in standard use⁴
• informal English⁵
• slang⁶
• a short form⁷
• written form of a reduction⁸
• dialectal⁹
• a verbal phrase representing casual pronunciation¹⁰
• relaxed pronunciation¹¹
• vulgar pronunciation¹²

The scholarly merits of each of these terms could be debated individually, but, the fact is, all of these terms have been used to one extent or another to categorize words such as gonna, coulda, oughta, and lotta – you betcha they have.