number of syllables in GIRL

I (US Mid-West) definitely pronounce this as having two syllables, with a schwa between the R and the L. In fact, I find it difficult to make a consonant cluster of RL. This is contradicted but dictionary entries that give the pronunciation as having one syllable.

How many syllables does the word have?

For many American English speakers, the distinction between /l/ and /əl/ is unclear when /l/ is in the coda of a syllable (it doesn't necessarily have to constitute the entire coda: see the related question about words like child, wild, and field) after a tense vowel, a diphthong, or a rhotic vowel. The exact set of vowels associated with this merger may vary between speakers; for example, I feel like I have a fairly robust distinction between /el/ (as in trail) and /eəl/ (as in betrayal) and /ol/ (as in stole) and /oəl/ (as in bestowal) although in fast speech I can imagine that I might merge them. (Normally, I pronounce /el/ as something like [eə̯ɫ] and [eəl] as something like [eɪ̯əɫ]). But I don't have a strong distinction between /oɪl/ and /oɪəl/: for me, it feels like oil basically rhymes with loyal.

This means that words that show this variation have an unclear number of syllables.

Wikipedia covers the possible environments in a section titled "Vile–vial merger" of the article "English-language vowel changes before historic /l/":

In some rhotic accents, [a merger may occur between] /-ˈɜːrl/ (girl, hurl, pearl, etc.) and /-ˈɜːrəl/ (referral), usually skewing towards two syllables.

In the standard General American accent, 'girl' has a single syllable, /gɚl/ or /gərl/.

But a large number of people separate 'rl'. This is similar to how the 'l' in bottle forms a syllable all by itself. This is called a syllabic l'. The sounds (which is not the same as letters) of 'r', 'm' and 'n' also can do that, be a syllable without the usual expected vowel as center of the syllable, though often the schwa 'ə' or 'uh' is considered the syllabic nucleus that comes with a syllabic consonant.

This is not an uncommon variant in all American varieties of English. It is not characteristic of any particular dialect. Some people in all varieties do it. That means that saying it does not pinpoint you to a particular accent, it is an acceptable variation in all accents.

It is not surprising that the consonant cluster 'rl' is sometimes separated. It is often a very difficult pair to pronounce by children and non-native speakers. Notice in the second clip, the announcer, who has a British English accent, gives two syllables for 'squirrel' which is perfectly natural alternative.

This is in contrast to the other direction. Some words, like 'fire' or 'desire', which in the standard pronunciation have two syllables for 'ire' = /aɪ ɚ/ or /aɪ ər/, can reduce to /ar/. It's not like the syllable distinction was emphasized in the standard pronunciation, but especially in the South or in the very similar AAE, they are sometimes non-rhotic (drop word final 'r's) and monophthongization (saying 'Ah' for 'I').