Learn Turkish: Fobim yüzünden kızarmaktan ve alay konusu olmaktan bıktım. - Because of my phobia, I am sick of blushing and being a subject of mockery.

The verb bıkmak (to be sick of) always requires its object to take the ablative case (-dan / -den). Because the things the speaker is sick of are actions, we take the dictionary form of the verbs (kızarmak and olmak) and add the ablative suffix. This creates the -maktan / -mekten structure you are learning, meaning "sick of blushing" and "sick of being."

Why use yüzünden instead of için for "because of"?

Yüzünden is specifically used for negative causes or when placing blame—it translates closer to "due to the fault of." Since a phobia causing you to be mocked is a negative thing, fobim yüzünden is the natural choice. You would use sayesinde for a positive cause, and için for a neutral reason.

How is the phrase alay konusu constructed?

It is an indefinite noun compound. It brings together alay (mockery/ridicule) and konu (subject/topic). To link them together to mean "subject of mockery," Turkish requires the third-person possessive suffix -su on the second word, giving us alay konusu.

The verb kızarmak means "to blush" here. Is it only used for people?