Trying to go to Turkey in spite of the visa ban between US and Turkey

UPDATE (as of December 2017) : The Turkish government is once again issuing e-visas (US$20) and visas-on-arrival (US$30) to US Citizens.

The details below are, at this point in time, incorrect but left for historic reference.

The other answer is wrong.

As stated in Timatic, the database used by airlines:

Visa required, except for Nationals of USA can obtain a visa on arrival for a maximum stay of 3 months. This does not apply to nationals of the USA departing from the USA and traveling directly to Turkey or transiting in a third country less than 24 hours.

So yes, if you make a stopover elsewhere for at least 24 hours (which you can prove using your boarding passes) you can get a visa on arrival.

If you're still not convinced, try to call the police at the airport and ask (+90 212 463 30 00). No guarantee they'll speak English though.

UPDATE: Timatic's been updated, now stating:

Visa required, except for Nationals of the USA can obtain a visa on arrival for a maximum stay of 3 months. They must be arriving from a country other than the USA and have a residence permit issued by the country they reside in.

So Americans residing in the US can no longer get a visa on arrival at all

No, of course not. Due to diplomatic disputes, the issuance of Turkish visas to US citizens was severely restricted on October 8th (and vice versa). Before that, US citizens could apply online for en electronic visa, which usually was granted immediately. Now, US citizens have to apply for their visa at a Turkish consulate or foreign representation and visas are only granted in very exceptional cases.

It is in this case irrelevant where you are flying from. As a matter of fact, without having a visa in advance, any airline would deny you boarding a flight to Istanbul, as you are not entitled to enter the country.

Edit: I tried to call Atatürk Airport to ask why the official information from the MFA and the current practice of the immigration police obviously do not match. The person I talked to at the airport did not want to or was not able to answer my questions, but asked me to call +90 312 2921000 instead, the number of the Ministry of Foreign Affairs in Ankara. There, I was connected with some kind of information desk, which pointed me to the official visa information page of the MFA and stated that e-visas and visa-on-arrival is currently not available to US citizens. They could not explain if or why visas are issued anyway by the immigration police at Atatürk Airport.