How to book a multi-day layover?

I just checked London to Istanbul and see very inexpensive fares. I didn’t check any other cities. Just google flights for options.

You will likely need to book your ticket “Open Jaw”, Portland (I assume) to Amsterdam and then Istanbul back home. I am not aware of any legacy carriers allowing layovers longer than 24 hours. Delta is my home airline and in their search, you can set your preferred layover time from 0 to 24 hours, however, I rarely see layover options longer than 8 hours. One thing to keep in mind is that once you get to Europe, flying between cities is cheap (even one-way fares). Amsterdam is one of our gateway cities from MPLS, so we often do a stopover of between 1 or 2 nights on each end of a trip. Just keep in mind that you won’t be able to combine your ‘to and from’ Europe flight segments to your inter-Europe flights, so give yourselves some leeway if flights get delayed (or cancelled).

What you're describing is the "multi-city" option on the airline website.
We used it frequently for business travel back in the day and it worked very well for us.
The real question for you will be whether it'll be any cheaper than doing your trip as an open jaw or by making liberal use of the various low-cost airline options available these days. Might be cheapest, for example, to book your trip as a RT to/from London (or another intermediate point), delay a few days, then use Easyjet or some other low cost option to get to your final (or another intermediary) destination. You'll need to make trial bookings in order to compare relative prices and schedules and to determine what's best in your particular case.
There was a time when using the multi-city option was a money saver but I'm not sure that's the case any more with the advent of the low cost carriers. .
In our particular case we've started booking a BA non-stop to London from Denver, delaying there for a day or two to rest and recharge, then proceeding on by using one of the discount carriers. It's an option that works well for us, but then we're retired and so don't have the time constraints on our vacations that we used to.
Doing a quick comparative search on Google flights for your proposed itinerary, I see relatively low-cost (one stop) flights to London from Portland, then very reasonable non-stops from there to Istanbul and return. Looks pretty convenient at first glance.
Bottom line is that it's doable, but it will take some research on your part to determine which of the options is best for you.

I'm also not aware of layovers on other airlines that exceed 24 hours. If you do a "multi city" search on Google Flights, you can add as many segments as you want, and choose the dates you want to fly (use the "add flight" button to add segments). So for example you could search PDX-AMS, AMS-IST, and IST-PDX. The search results will note if you have to buy separate tickets for the segments.

However, I picked a random set of dates in May/June and there were plenty of options that involved just one stop from PDX, including one via San Francisco (Turkish Airlines PDX-SFO-IST, about 17 hours total) and also via Amsterdam (KLM/Delta 16.5 hours total). Return flights have less options, but there is a KLM/Delta flight from Istanbul that has a 14 hour overnight layover in AMS. None of them are "cheap" flights, but you're unlikely to pay much less than $1200 per ticket unless you're going in very off season (my random date search showed tickets in late May at $1150, which is actually quite good for PDX prices). Note that British Airways starts flying from PDX next June, that may open up some options if you're going after then.

Good luck!

We have been to Turkey from SFO twice on Turkish Airlines. It was a non stop flight. Not sure if these flights operate every day. Also don't know which airport you want to use, but if you can use SFO, you may want to check this option.

Most airlines do not allow lay overs of more than 24 hrs. Icelandic uses the extended layover as a marketing tool. But if you want to do a longer lay over, you have to book it as multi-leg (multi-city) flight on basically one ticket. So SF to Amsterdam or London, on to Turkey, and home. We have done flights with as many as five legs on one ticket.

The non-stop flight from SFO on Turkish Air would appear be your best option. The crossing of 10 time zones is going to be something of an ordeal no matter how you cut it so there's a lot to be said for just getting it over with by booking the direct flight.
It would require seperate tickets in order to connect to the Turkish Air flight, so depending on how risk averse you are you might want to fly down to SFO the day before to ensure that you're in place for the international flight.

Terminology - layover vs multi-city flights.

A "layover" is a connection between different cities/airports and is less than 24 hours. If you checked baggage, the baggage would be sent to your final destination and not be available during the layover.

A "multi-city" is a ticket allowing you to stop over at one or more destinations. Checked baggage would be unloaded and available at each multi-city destination. The multi-city itinerary could be a round trip. A multi-city itinerary I used a number of years ago:

Departed hometown USA connecting in Frankfort to Prague (first stop). Second leg was a flight to Oslo (second stop) connecting in CDG. Third leg was returning from OSL.

How to book: https://www.airtreks.com/how-to-book-multi-city-flight/

Air Canada offers stopovers in the same way and for the same reason that IcelandAir does - to encourage you to spend tourist dollars in their countries. With the Air Canada stopover program, you can build in a stopover inside Canada. With IcelandAir, you can have a stopover in Iceland, not anywhere on their route map. I don't think a few nights in Montreal is what you're looking for in this case. But check other airline websites for "stopover" and see if anything turns up. It definitely exists.

For reference in your search, not because I think this is what you want, here is a description of the AC program: Air Canada Stopovers

ETA: just found this: 9 Airlines That Offer Free Stopovers In Amazing Cities
Air Canada.
Iberia.
Qatar Airways.
Finnair.
Ethiopian.
Emirates.
Icelandair.
Etihad.
Singapore Airlines.

Probably irrelevant for the OPs trip, but TAP air Portugal allows multi-day stopovers in Lisbon and Porto. You can book them easily on their website.

If no airline offers an official multi-day layover then book home - London (or wherever) and Istanbul- home one ticket using the multi city option (on the website, sometimes you have to search a bit to find it). Then book a separate, cheap flight from London (or wherever) to Istanbul. It may take trying a few options before finding something reasonably priced but it should be doable for not much more than a round trip.

I agree with some of the previous posters, short of finding a deal with a stay through one of the National Airlines, your options are two:

1: Book using the Multi-City option through an airline. Likely that will be more expensive than a Round Trip to Istanbul.

1. Book as two separate tickets, either through the same airline or mix it up. An Example would be Round trip to London corresponding to your start and end dates, then find a budget or other option from London to Istanbul Round Trip, giving you days in between flights. If you go this route, resist the urge to arrive in London and immediately leave (such as your return leg) treating it like a transfer, spend at least a night between the flights.

Thanks for all the great suggestions. I had a feeling that the multi-stop option, plus a cheap ticket within Europe would be the way to go. I’ll definitely look into the other airlines offering stopovers. And BA will start flying from PDX next summer, which opens up a lot of options.