English expression for what Germans call 'Kompetenzgerangel'

If I understand correctly, this is called a turf battle or turf war. Oxford Dictionaries Online defines turf war (also turf battle) as:

An acrimonious dispute between rival groups over territory or a particular sphere of influence.

I perceive a turf battle as less serious than a turf war. These terms are definitely used for disputes between government agencies and between divisions in a company.

If you want to make it sound immature or childish, you could use turf squabble, which isn't anywhere near as common as turf battle or turf war, but which I can find using Google search; everybody would understand what it means, and perceive it as less serious than a turf battle (which itself isn't always used for serious disputes).

Although a little uncouth:

Pissing contest

OED defines it as:

vulgar slang
A contest or rivalry in which the main concern of the parties involved is the conspicuous demonstration of superiority.

It is usually used in the context of unimportant matters/competencies.

I have head it used in polite circles and in business situations, but always in the pejorative sense.

One option that is a little bit less casual sounding is jurisdictional dispute.

From jurisdiction:

3. The extent or range of judicial or administrative power; the territory over which such power extends.

• OED

This term usually refers to the realm of legal or executive power, but OED's definition 2 indicates that it can refer to a dispute over spheres of power more generally.

2. Power or authority in general; administration, rule, control.

Power Play

power play 1. Tactics exhibiting or intended to increase a person's power or influence. ‘the sexual power play of their relationship’

fight for power is also translated as ger. Machtkampf. However power play might also mean "2. Offensive tactics in a team sport", especially in American English. It appears that the childish connotation of play is reduced to actually fair games, whereas the term power struggle might embrace a serious undertone.