Learn Turkish: Polisin baskını üzerine şüphelilerin gözaltı süresi uzadı. - Upon the police's raid, the suspects' custody period was extended.

The postposition üzerine (meaning "upon" or "as a result of") is following a specific, defined event: "the police's raid". This requires a standard genitive-possessive construction. Polis takes the genitive -in, and baskın (raid) takes the possessive -ı, forming polisin baskını. The postposition üzerine then immediately follows this completed phrase to show the sequence of events.

The English translation says "was extended" (passive), but uzadı is active. Why didn't we use a passive verb?

The verb uzamak is an intransitive verb meaning "to get longer" or "to extend". In Turkish, it's very natural to say a period of time "got longer" on its own (süresi uzadı). While you could use the passive causative uzatıldı (was deliberately lengthened by a judge or authority), uzadı is perfectly standard here to focus on the change in the duration itself rather than the person who changed it.

How is the noun phrase şüphelilerin gözaltı süresi linked together?

This is a nested noun chain. First, we have the compound gözaltı süresi (custody period), where süre takes the possessive suffix -(s)i. To say whose custody period it is, we add şüpheliler (suspects) in the genitive case (-in). The single possessive suffix on süresi works double-duty for both the inner concept ("custody period") and the outer ownership ("the suspects' period").