I am trying to find the cheapest one-way fare between Europe (including IST) and North America. So far, I have only found IST-YUL on certain days of the week for US$345 (September and October).

I have the following flights booked (adjusted some flights for two trips booked before the promotion was announced):

DFW-IST-ATH (TK) $385 (W class - 25% miles)
Mediterranean Cruise ATH-BCN (already planned before promotion announcement)
BCN-MAD (AVE) $35
MAD-IAD (IB A321XLR J class 50K Avios and $140 already booked before promotion announcement)
JFK-SIN $520 (SQ nonstop in R class booked before promotion announcement)
SIN-IST-DXB (TK) $697 (Q class - 100% miles)
DXB-IST-CAI (TK) $267 (U class - 25% miles)
CAI-IST-SYD (TK) $740 (Q class - 100% miles)
SYD-LAX (Staff Travel) $93
**All fares in USD including taxes and fees**
TK Status Miles for entire itinerary:29,537 miles (Currently TK Elite using status miles to protect status and possibly get to TK Elite+)