Some corrections to what's posted above:
Turkish Air will fly you to either San Fran or LAX for a direct flight to Istanbul...
I'm assuming your "Mayberry-on-the-Willamette" reference is Portland (I thought Eugene was Mayberry-on-the-Willamette?).
In either case, your nearest connecting city would probably be SEA. TK currently has one nonstop between SEA and IST every day, next year they're adding a second flight on some days (great news for SEA-based TK fans). Turkish Airlines also currently has nonstop flights between Istanbul and the following other cities in the USA: JFK, BOS, MIA, ATL, IAH, DFW, EWR, ORD, LAX, IAD and DTW (also in Canada: YVR, YUL, YYZ). They are expanding regularly, look for new cities served (I believe DEN is coming next).
It's an 11-12 hour flight (at least from Seattle, where I live), not 15. Flight length varies by season and weather (jet stream shifts). It's a long flight in any case, though if you're in business class (which you seem to be stalking) the time really will pass quickly and easily.
Turkish Airlines does not partner with Alaska Airlines in any way. Alaska & Turkish do not "interline" bags (so you can not check a bag with Alaska Airlines in PDX through to IST).
United Airlines is a Turkish partner. So (assuming you have a single ticket all the way), you should be able to check a bag with United in PXD thru to IST and beyond to your final destination; if you can find a United flight from EUG to SEA, the same should be possible.
On a return flight from IST you would clear passport control and customs at your first US airport, so presumably SEA.
@ thenosbigs - OP is not planning a self-inflicted 17-hour redeye blitz visit to Istanbul after peeling themselves off a plane. They are considering the free city-tour benefit that this airline offers all passengers with a long layover who qualify - they are going to be in Istanbul for a long layover, the question is simply how best to use that time. It's a v-e-r-y quick, superficial, fully-guided-tour snapshot of this huge, complicated city (so not requiring much metal acuity or energy). I assume the OP is interested in this as a "freebie" benefit that just happens to come with their ticket, rather than just spending those 17 hours cramming food in the airport lounge (though I have to admit, it's a very nice airport lounge, with lots of great food - try the lentil soup, it's sublime!). Passengers on this free tour don't need to worry about making any arrangements (other than signing up and showing up for the tour itself), just get on the tour and follow instructions, they drive you back and forth. You would need to get the Turkish e-Visa (and I'd want to be sure I had enough time to make my onward connecting flight after the tour), but they run these tours twice a day, every day, so they know all about the bad traffic and how much time things take.
Yes, there will be some jetlag but by flying the long overnight leg in business class (in a quite comfortable lay-flat seat), I have found that I get a good night's sleep, and arrive in IST nicely rested and ready to hit the ground running. When you do finally reach your actual destination, you may be ready to hit the sack for a good long sleep, though (what's your final destination? It could make a difference in what kind of shape you're in there - ie headed to Venice, or Johannesburg...?).
I have friends who just booked a flight in TK business class from SEA to IST then after a long layover there, are connecting to Milan, and they may opt to do the free city tour (or they may just opt to relax in the lounge, catch a shower, get a massage, eat, etc.).
It's certainly a cool option if you are in the mood for it. I'd certainly consider it. Have fun.