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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 you can get a visa on arrival at the airport visa desk if not arriving directly from the US, and having spent at least 24 hours transiting a third country. In this case, you can visit Istanbul.

Otherwise, you will be refused entry, and cannot visit Istanbul, since no visas or e-visas in advance are being issued to US citizens

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 are arriving from.

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

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

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

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 you can get a visa on arrival at the airport visa desk if not arriving directly from the US, and having spent at least 24 hours transiting a third country. In this case, you can visit Istanbul.

Otherwise, you will be refused entry, and cannot visit Istanbul, since no visas or e-visas in advance are being issued to US citizens

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 are arriving from.

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

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 you can get a visa on arrival at the airport visa desk if not arriving directly from the US, and having spent at least 24 hours transiting a third country. In this case, you can visit Istanbul.

Otherwise, you will be refused entry, and cannot visit Istanbul, since no visas or e-visas in advance are being issued to US citizens

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 you can get a visa on arrival at the airport visa desk if not arriving directly from the US, and having spent at least 24 hours transiting a third country. In this case, you can visit Istanbul.

Otherwise, you will be refused entry, and cannot visit Istanbul, since no visas or e-visas in advance are being issued to US citizens

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 are arriving from.

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

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 you can get a visa on arrival at the airport visa desk if not arriving directly from the US, and having spent at least 24 hours transiting a third country

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 you can get a visa on arrival at the airport visa desk if not arriving directly from the US, and having spent at least 24 hours transiting a third country. In this case, you can visit Istanbul.

Otherwise, you will be refused entry, and cannot visit Istanbul, since no visas or e-visas in advance are being issued to US citizens