What do you call the person in charge of a district?

Mukhtar is a word in English too (although I've never heard it before):

(in Turkey and some Arab countries) the head of local government of a town or village.

_{Oxford Dictionaries}

The responsibilities of a mukhtar are unique; there's nothing exactly the same in the west (which is why we borrowed the term).

I'm not an expert in law (especially Middle Eastern law), so this Turkish site helped to explain some of their duties:

Villages

The mukhtar is both the representative of the village and symbolizes the state in the village. The Village Law ... imposes significant functions and responsibilities on the mukhtar as well as equips him/her with very important powers. The mukhtar plays an active role in ensuring the security of the village and procuring all public services. S/he facilitates the work of central government agencies.

Neighborhoods

The neighbourhood mukhtar serves as a bridging link between the neighbourhood residences and public bodies including particularly the municipality. The mukhtar discharges such significant functions as identifying the poor and provision of assistance, renewing voter registers, informing the relevant agencies of problems and failures in services of education, health, security and sanitation etc.

In my experience, these duties are handled by the government (spread through multiple departments and entities it subcontracts with). On the other hand, in the US at least, nobody needs to help prove your residency (if you don't have something like a driver's license, your mail will work instead).

Of course, there isn't one "system in the west", and there are various differences between each one. (You can often find different systems inside sovereignties themselves, such as state law in the US.)

I would translate mukhtar as chieftain:

1. the chief of a clan or a tribe.
2. a leader of a group, band, etc.: the robbers' chieftain.

Historical Examples

During a formal visit to the chieftain of the tribe, he was offered tea.
The Soul of a Child Edwin Bjorkman

The chieftain gave his orders for the defense of the village.
Despoilers of the Golden Empire Gordon Randall Garrett

http://www.dictionary.com/browse/chieftain