The name "baba ghanouj" is Arabic, and is written differently depending on how you convert the Arabic script into English. Sometimes it is written with an "sh" instead of a "j".
The final consonant in "baba ghanouj" is ج ; in Arabic, this is usually pronounced [dʒ] (as in the English "j" sound) or [ʒ] (like "Jacques"). (In Egypt, this even gets a [g] pronunciation.) So, the "j" spelling of "baba ghanouj" reflects a certain standard transliteration of Arabic spelling. For some reason, the [dʒ] or [ʒ] was devoiced (a.k.a. neutralized) into [ʃ]. However, this is not normal in the common Arabic dialects that I am familiar with, so I seriously doubt that Arabic has anything to do with the cause.
In any case, final consonant devoicing occurs in a number of languages as a broad and general process, and occurs in many more languages in one-off instances from time to time. So, either the word originally came into English via another language that neutralized the final consonant, or we neutralized the final consonant ourselves. The most likely candidate to take an Arabic word, neutralize the final consonant and then pass it on to English? Turkish. (This is my educated guess.)
Short version: for some reason, the sound became devoiced, but the spelling with a "j" reflects Arabic spelling.
So which pronunciation is standard for the [ʊ] sound? Rounded or unrounded?
Certainly there is some rounding, but because roundedness is not phonemic in this position, there is also considerable variation in how much of it actually occurs in any given word and speaker.
For example, you will find that it is generally somewhat more rounded in pull and full than it is in put and foot respectively. That’s because having an r or an l right next to it rounds it off a bit — which is why it is a bit more rounded in root and rook than it is foot or cook. Same with rookie versus cookie, where the first version is a bit more rounded than the second. And of course, a w helps: compare how wool is even more rounded than full, and also moreso that wood.
I believe English has no words with [ʊw], as that seems redundant. However, it can occur in phrases, especially in some dialects, where something like I knew it full-well may approach that.
However, it is still perceived as the very same phoneme in all those words and cases I’ve just listed above.
Correction — or not
I said that I thought English had no words with [ʊw] in them. And at the end of the day, I still believe that. However, I have discovered that grepping the OED yields the apparent existence-proof counterexample of Rauwiloid, which means:
A proprietary name for a hypotensive preparation containing a number of alkaloids extracted from Rauvolfia serpentina.
You also have compound words whose first element ends in [aʊ] (rather than [aw], as it is sometimes spelled) connecting to something that begins with [w], and which have in effect a “double w” in them, you expand the list to include such things as:
bow-wow, powwow, skeow-ways, wow-wow
Finally, if you consider the sound in words like no and micro to be
an [oʊ] diphthong rather than [ow], then you get all these, most of which were originally compounds of some sort:
froward, frowardly, frowardness, glow-worm, Holloway,
hollowwort, Howeitat, Khowar, meadow-wink, microwave, microweld,
Moldo-Wallachian, nowise, Oldowan, Parowax, powan,
shalloway, slow-worm, swallowwort, werowance,
yellow-wood, yeowoman.
For example, yeowoman theoretically yields /ˈjoʊwʊmən/, at least in North America. Still, there is a reasonably convincing argument to be made that that one is better written as simply /ˈjowʊmən/.
Slightly less uncommon is nowise, which is a compound of one word ending in a diphthong connected to another starting with a triphthong, so /ˈnoʊˌwaɪz/.
But I am still highly dubious of the existence of [ʊw], because I think it fuses into the semi-consonantal glide, [w]. After all, nowise and no eyes are homophonic, so I think this idea of [ʊw] is very hard to justify, and so I stand by my initial statement.
Even towel is usually pronounced with just one syllable, /taʊl/, thereby rhyming with cowl /kaʊl/. Even with folks who work very hard to put two syllables into that, with /ˈtaʊ.wəl/, I submit that you could write that /ˈtawːəl/ and avoid the whole controversy of whether a semi-vowel/semi-consonant/off-glide is really /ʊ/ or really /w/. However you write it, it seems like the same sound to me, such that bisyllabic towel just has a geminate [w]: /ˈtaw.wəl/.
Best Answer
I would pronounce "ne" (all by itself) as close to "neh". It's also "neh" in "necessary", "negligent", "Nellie", "nepotism", "nest", "nettle", "never", "nexus", -- but "new" is closer to "neew".
As the others have pointed out, English spelling is more like a suggestion about pronunciation. I think this is because we've borrowed from so many other languages - French, German, Italian, Russian, British English .....
One of the most famous examples is "through", "tough", "cough", "though", "rough", "thought", "bough". That's an extreme case - but I don't think it's anywhere nearly as complex as tonal languages like Chinese.