No matter how they start off, topics about motor oil invariably turn emotional. Fascinating
Just to address something said earlier about 5W20 being generally full synthetic or synthetic blend, I checked around quickly. Though this may have been the case some years ago (I recall this as well), today we have quite a few conventional 5W20s out there.
One possibly important detail hasn't been discussed here yet : the Chrysler MS6395 certification. Motor oil that goes into our Cherokees *should* meet this certification, according to our beloved manual.
I found this nice write-up, for 5W20 : http://jeepoffroadadventures.com/wp/?page_id=787
Further searching led me to find that Mobil 1 Super Synthetic now meets Chrysler MS6395
Searching for 0W20 (for the Tigershark engine) yields fewer results, and most seem to be blends or full synthetics.
Dealerships usually purchase motor oil in bulk, and unless told otherwise, I am convinced they go with the cheapest fluids they can buy, meaning not a full synthetic or even a blend. 5W20 is now very popular so surely available in conventional bulk. Ask your service person and they should tell you what - exactly- they put in your engine.
For 0W20, they may carry bottles because of the rarity and chances are it would be a high quality oil (blend or full syn).
Anecdote : our FCA dealership lets customers supply their own oil for oil changes if they so desire. I've tried this at various shops and have been told *NO* because they make a living from profits on parts. I've even offered to let them charge me a full conventional oil change but use my jug of synthetic to satisfy my OCD (haha), but they refused. Needless to say I take my business elsewhere, or do it myself when possible. Most shops I see now charge an extra $30-$40 for synthetic, which is totally ridiculous and, quite frankly, insulting when you know what oil sells for... [/rant mode]
Anyhoots, enjoy your Jeeps and your oil