The Turian society is highly militarized and based on a meritocracy. Throughout their lives, Turians are promoted and demoted throughout 27 "citizenship tiers" by proving their worth and value to their society. Primarch is the highest rank you can attain.
The Codex entry on Turian government says the following:
At the top are the Primarchs, who each rule a colonization cluster. The Primarchs vote on matters of national import. They otherwise maintain a "hands-off" policy, trusting the citizens on each level below them to do their jobs competently.
I'm finding difficulty with an analogue to existing governments because I'm not sure how large the scope of "colonization cluster" means, whether it's small colonies on the planet itself or planetside colonies as a whole.
Considering the import of how necessary it was for them to figure out the next in line of succession, and that Primarch Victus is able to speak for the Turians during the meeting between the Krogans and Salarians, I'm leaning towards the latter, in which case I don't think we currently have an exact analogue of someone who speaks for the entire planet.
However, if you think of the separate colonies of the Turians as states, then you could roughly think of the Primarchs as Senators (in regards to the U.S. government) who vote on matters of import for the colonies/planets they represent. This analogy is a little flawed though because given that Palaven is the Turian homeworld, the Primach seems to hold a bit more sway in that he's allowed to represent the race as a whole.
In short, there's not a directly analogous position given the scale of power, but we can approximate.