I'll probably be toasting my bum in coach if I do take TK up (scary). I will second those who find this challenge less entertaining than SAS' for the lack of variation and the ease of brute forcing with cash. Nevertheless, I do appreciate these challenges. I did a little research, and here are my thoughts so far:

Stopovers and nesting itineraries can help reduce positioning costs significantly. TK has a very generous stopover policy, with 4 free stopovers offered on most itineraries. Find fare rules on Expedia.
Asia/Africa/Europe is better done as part of a connecting itinerary to the other 3 continents. Personally would pair Africa with Oceania (Egypt to Australia), Europe with South America (VIE-BOG) and Asia to North America (MCT-IAD) at this stage. This rounds out to approximately £2000 in economy.

Northern Africa to South America is quite tough on TK (for reasons discussed upthread)
African routes outside of Northern Africa, ADD, and CPT are largely poor value for the purpose of this challenge

(For Asia) Fare to and from KUL, SIN, HKG, ICN seem to be more generous for travel to Europe, North America, and Egypt. Fares at DXB are great for the challenge, but questionable value for distance flown. Fares to India are very competitive.
(For Europe) Fares at FCO, MXP, BLQ, MAD, VIE, ARN, GOT, BUD, VNO, MAD, ATH, and AMS are noteworthy for travel from Oceania, to South America, and both ways for North America.
(For North America) Fares at DTW, DEN, ORD, NYC, WAS offer better value, but one-ways can be 80% of a return.

OSL-HAV is available for approximately £500. Would be quite an interesting way to check off North America.

Good luck to those attempting!