Learn Turkish: Kazayı kıl payı atlattık, yoksa ölecektik. - We narrowly dodged the accident, otherwise we would have died.

It literally translates to "hair share" (kıl means a single hair, and pay means share or portion). Turkish uses this idiom the exact same way English uses "by a hair" or "narrowly" to describe barely avoiding something.

Why does kaza (accident) take the accusative case here as kazayı?

It is the direct object of the verb atlatmak (to dodge, get over, or survive). While English sometimes uses prepositions like "escape from," Turkish treats the accident as the direct target of the dodging, so it requires the specific direct object marker -(y)ı.

How does ölecektik translate to "we would have died"?

This verb combines the future tense suffix -ecek with the past tense suffix -ti (plus the "we" ending -k). This "future in the past" structure literally means "we were going to die," and it is the standard way Turkish expresses "would have" for a consequence that was about to happen but ultimately didn't.

This lesson focuses on iyi ki. How could we add it to this sentence?

You could naturally say, İyi ki kazayı kıl payı atlattık, yoksa ölecektik (Thankfully we narrowly dodged the accident, otherwise we would have died). Adding the evaluative marker iyi ki at the beginning is a great way to express your relief about the completed action (atlattık).