Collounsbury and Tamerlane have already covered the OP question quite well.
The southwestern or Oghuz branch of the Turkic languages includes Turkish (by Turkish I mean the language of the Republic of Turkey…Istanbul…Ankara…what used to be called Ottoman Turkish), Azerbaijani, Türkmen, and Gagauz (the last-named language is spoken in Romania). These languages are quite closely related to each other. Turkish and Azerbaijani have considerable mutual intelligibility between them (I saw one quantitative study that put the figure as high as 70%), Türkmen less so.
The Oghuz peoples migrated from Central Asia toward the Middle East in the 10th century. The Oghuz soldiers who wound up in Iraq were hired as the praetorian guard of the ‘Abbasid caliphs, and eventually came to exercise the real power in the state with the caliph as figurehead. A later branch of these peoples established the Seljuk sultanate in Anatolia, and a spinoff of the Seljuks later became the Ottoman state.
The Azeri language is called “Turkish” by the Iranians.
Türkmen nomad tribes continue to wander in Türkmenistan itself, in Iran, northern Iraq, and in Turkey. To my understanding, they are all more or less the same subgroup of the Oghuz peoples; I imagine those in Iraq speak a somewhat different dialect of essentially the same Türkmen language, possibly a bit closer to Azerbaijani and Turkish, with a lot more Arabic loanwords. The Türkmen tribes who wander Anatolia are known by a different name: Yörük.
That’s the whole point, Collounsbury. There are Lumbers and there are Splitters (Yes, those are actually the technical terms). What one person considers to be a dialect is what another considers to be a language. Some folks even go so far as to say that any difference between the speech is another dialect, instead of merely a regional word usage or definition.
Essentially, there’s no way to get around the subjectiveness displayed on that site. If they were to do it another way, then another group would say that they’re being subjective also, but in a different way.
I am well aware of the above my dear fellow, well aware.
Well, that was not at all in dispute, I was, however, taking learned exception to the strange inconsistencies in the SIL standards, and to the degree to which they split some but not all languages. Thus, once again, my reference to the extreme degree to which they slice and dice Arabic, while similar levels of difference in Euro langauges are not sliced. Split as you like, but do it consistently. I also was drawing attention to the fact that indeed they do seem to be, when it suits them, extreme splitters.
The inconsistencies in the Ethnologue is probably because it is both a top-down and bottom-up project that is specifically driven by Protestant missionary work. Thus, languages of areas that might be targeted may end up being given more detail than “English”.