Why are people under the impression that the X-wing has, or ever had, a jet engine?

As far as I'm aware, there's been no on-screen demonstration of any rotor in those tubes. I always assumed their were either forward-facing radiators for the engines, or some kind of intake for radiation/space debris for whatever reason.

The TIE/FOs are a new model, but don't have the cogs seen on the Special Forces TIEs.

I just realized how ridiculous the new X-Wings are. They have what look like Jet Turbines, but with the wings open, each turbine is split in half and couldn't possibly function.

Jet Turbines wouldn't function in space anyways, so there's that.

Actually if you look at this more recent image:

http://photos.vanityfair.com/2015/05/01/5543ca9adb753b82389cbdc6_vanity-fair-star-wars-03.jpg

the engine turbines look like the design is a bit different. Also, the hull on the above x wing is lighter in color..less grey and closer to white. But the turbines definitely don't look like they will be rotating in that pic. It seems there might be some slight variation among the t-70 x wings.

Not a variation of engine design, just simplified details on a miniature representation of the life-size T-70.

The designers of the T-70 took a lot of cues from Ralph McQuarrie's original concept art so if anyone is to blame its him. Anyway I never saw the T-65 "turbine engines" as turbine engines, at least not in any real-world sense of the word, and to my knowledge all the Star Wars material out there doesn't describe them as such, so likely no rotating parts in them, which means its shape doesn't really matter.