There probably aren't that many people who are well-versed in the upgrade schemes of multiple foreign airlines here who can provide you with a good comparison.
Generally, mileage upgrades are easier and cheaper on US-based carriers. I think DL is the only US carrier to fly to IST. DL requires a minimum "M" fare class to upgrade, which will likely be at least $3,000 r/t. You can transfer Amex MR points to DL Skymiles.
Other than that, you may have to do some of your own research.
Here's what I would look for, on each airline's respective FFP website or Amex/SPG:
1) How many miles to upgrade
2) Can I transfer miles into the program and what's the transfer ratio
3) Is there a minimum fare required to upgrade
4) Is there a co-pay in addition to the miles required for upgrade (common for US-based carriers other than DL), and if so, how much. Is it fixed (a la AA) or sliding based on fare class (a la CO/UA).
The gating item may be #3 for a lot of the Euro programs (and DL). A (pay) site like ExpertFlyer can help you identify how much a particular fare class will cost.
And of course something to consider is getting "close" to IST on a US-based carrier (e.g., BUD on AA, ATH on CO though it may be seasonal) and then just flying coach the rest of the way. As it relates to this, keep in mind that most if not all flights from other European gateways to IST will have European seating, which in business class generally means a regular coach seat with a few extra inches of legroom and the middle seat empty. So you'd want to minimize intra-european flying regardless of class of service -- connecting in FRA or ZRH would be better than LHR, for example.
Generally speaking, upgrading international coach tickets is not a cheap proposition. Generally $700-1500 more than the lowest available coach ticket. Many find redeeming a business class award to be a better value proposition. It should only be about 100k miles r/t in any of the major US-based FFPs.
You might also just consider purchasing Comfort Class (Premium Economy) on Turkish, which I think is supposed to be pretty good -- like old-style (mid-90s) business class.