Learn Turkish: İki gündür uykusuzum. - I have been sleepless for two days.

The suffix -DIr means "for" a period of time. Like most Turkish suffixes, it changes to follow vowel harmony. Since the only vowel in gün (day) is ü, the suffix takes the matching rounded form -dür.

Why does the translation say "I have been" if there is no past tense suffix in the sentence?

This is a key difference between English and Turkish! When you use the duration suffix -DIr for an ongoing situation, Turkish uses the present tense. While English uses the present perfect ("I have been"), Turkish literally just says "For two days, I am sleepless."

How is the word uykusuzum built?

It is made of three pieces: uyku (sleep) + -suz (without / -less) + -um (I am). Put together, it literally means "I am sleepless".

Why did the personal suffix "I am" become -um instead of -im here?

It is a chain reaction of 4-way vowel harmony. First, the "without" suffix became -suz to match the u in uyku. Then, the "I am" suffix looks at the last vowel before it, which is now the u in -suz. To match it, the personal suffix becomes -um.