Hello old friends and millionaires from SK!
It is a shame TK changed the terms so abruptly. I hope you have all booked your tickets and are ready to go. I was weighing up my routes before bringing forward my shopping schedule when I came across the bad news. In hopes of it reopening again, I'll continue from my previous post and share what my research has yielded before being abruptly interrupted here:
TK is quite particular about open-jaws and stopovers, contrary to my initial findings, and many deeply discounted fares are ineligible for a free Istanbul stopover. This manifests in the form of drastically inflated prices on stopover itineraries against connecting itineraries, where all constituent flights show up at the discounted rate in other itineraries. That is the most straightforward way to verify that a deeply discounted fare is ineligible, save for reading the actual fare rules that do not show up on itamatrix reliably, or having an Expert Flyer (EF) subscription.
Contrary to some suggestions upthread, my experience suggests married segments are not an issue on TK. What does occur, however, is that one segment in a connecting itinerary sells out of a lower fare class, and therefore the entire connecting flight (or technically fare component) prices out at a higher fare class. This can be mitigated by using the multi-city tool wisely, especially via Google Flights (GF). This is because TK's own booking engine seems to be very frugal with computing resources, and does not show all options where a significant number of them exists. Instead, it shows options by departure time, direct options first, and the more segments you have, the earlier the list cuts off. If fare is found on GF but not TK, try a TA or if need be, an OTA.
While married segments have not been found in my research, TK does offer round-trip fares that can range from a small discount on the one-way to being so significant it is lower than the actual one-way fare being sold. To leverage this while trying to hit continents, one can book open-jaws to have a cheaper fare component tacked onto a more expensive one. Do note that combinability issues may arise, especially with deeply discounted fare classes, but the general idea stands. Open-jaws on TK can be restrictive in that the open-jaw on the destination side may be restricted to regions (e.g. Middle East+North America, South East Asia etc.) which might result in smaller jaws being impermissible while bigger jaws tickets just fine.
I personally found success with [KUL-CAI, (any Egyptian port)-(any Australian port)], which saved me from having to do a return to/from Australia, and the KUL-CAI section being just $250 also helped me save a couple coins.
I have also found widespread availability for Europe-North America between SKP/PRN-NYC, available for around $500, absolutely a steal if not for SK (oh good ol’ SK our old friend) also plying the same route on some dates for just $300.
As others have noted, ATH-GRU is decently priced and its availability is deceptively low due to ATH-IST being in high demand. Where a fairly priced ATH-IST segment is found, IST-GRU can be easily had by adding a stopover at IST and trying a few dates, and the return is even easier, for about $650. Other noteworthy finds for deeply discounted return fares include BOG-FCO and ZRH for about $750, SSH-YUL and MCT-IAD at about the same, and ORD-(South East Asia port 1),(South East Asia port 2)-LAX/SEA/SFO for about $800. Also, IST-PRN/SKP for under $50 one-way as discussed upthread.
I hope this could become of use to those of use who needs more tickets to complete the challenge, and that TK changes its mind back. To the rest of us, I look forward to seeing you on the road!