Learn Turkish: Genç kadın tökezledi ve tabağı düşüreyazdı. - The young woman stumbled and almost dropped the plate.

This is the approximative compound verb in action. We take the verb root düşür- (drop) and add the suffix -eyazmak, which means "to almost do" or "to come close to doing" something. Finally, we add the simple past tense -dı, giving us düşüreyazdı (almost dropped).

Could we just say düşeyazdı instead of düşüreyazdı?

No, because that changes the meaning entirely. Düşmek means "to fall," so düşeyazdı would mean she almost fell herself. Here, she almost dropped the plate, so we must use the causative verb düşürmek (to make fall / to drop), resulting in düşüreyazdı.

Why is the word for plate written as tabağı here?

It takes the specific accusative case suffix -ı because "the plate" is the direct, specific object of the verb. When a vowel suffix is added to a noun ending in -k, like tabak, consonant mutation occurs and the k softens into a ğ.

Is the -(y)eyazmak structure common in everyday spoken Turkish?