eksedziĝi

Three pieces, read left to right. Eks- is former. Edzo is a spouse - a husband, with edzino for a wife. -Iĝ- is the suffix that means to become. Stack them and you have "to become a former spouse", which is precisely and unsentimentally what getting divorced is. Nothing in the word is borrowed, nothing is idiomatic, and a learner who has met all three pieces separately can build it themselves without ever having seen it. This is the language's central claim, demonstrated on a word English needs a Latin loan for.

Get the direction right: -iĝ-, not -ig-. Eksedziĝi is what happens to you - it is intransitive and it takes no object. Ili eksedziĝis, they got divorced. Eksedzigi, with -ig-, is to divorce somebody else, to make them a former spouse, which is what a court does: la juĝisto eksedzigis ilin. One letter, opposite roles, and if the igi/iĝi pair still feels slippery this is the word to fix it on, because the two readings are so obviously different in the world.

And the pair with divorco, rank 1,896. Divorco is the noun, the borrowed one, and it is the legal term: the decree, the proceedings, the fact on the record. Eksedziĝo is the same event named in Esperanto's own materials, and it is the ordinary word. Both are correct. Use divorco for the paperwork and eksedziĝi for the life. The verb divorci also exists and is used, chiefly by people who find the three-piece build a mouthful.

A word on register. There is no coyness in eksedziĝi and no sneer either. Esperanto has no equivalent of the euphemisms English reaches for, and the plain word is the kind one - it states what happened and stops. The same construction runs through the family: eksedzo an ex-husband, eksedzino an ex-wife, and eks- will attach as readily to prezidanto, instruisto or membro when you need it.