Learn Hardcore Turkish: Şehir parkını iyileştiriyorum.

That -ı is the accusative case marker, showing şehir parkı is a definite/direct object (“the city park”). Turkish marks definite objects with -ı/-i/-u/-ü (chosen by vowel harmony). Since park has a (a back vowel), the accusative form is park-ı (plus a buffer, see next question).

What’s the extra -n- doing in parkını?

When a word ends in a vowel and you attach a suffix starting with a vowel, Turkish inserts a buffer consonant to avoid a vowel clash. Here parkı ends with ı, so to add the accusative -ı again we insert n: parkı + n + ı = parkını.

Can you break down iyileştiriyorum into its parts?

Sure. iyileştiriyorum =
1) iyileştir- (verb stem “improve”/“make better”)
2) -iyor- (present continuous tense, “is/am/are …-ing”)
3) -um (1st person singular suffix “I”)
Putting them together gives “I am improving.”

Why isn’t the verb iyileşiyorum instead?

iyileşmek means “to get better” or “recover” (intransitive: no object). iyileştirmek is its causative/transitive form meaning “to make something better.” Since you (the subject) are actively improving the park (an object), you need iyileştir-, not iyileş-.

Why is there no word for “I” in the sentence?

Turkish is a pro-drop (null-subject) language: the subject pronoun is often omitted because -um on iyileştiriyorum already tells you the subject is “I.” Adding ben (“I”) is possible for emphasis but not required.

Why does şehir have no suffix—why isn’t it şehr-i or şehir-in?

Here şehir parkı is a compound noun meaning “city park.” In noun–noun compounds, the first noun usually stays in its bare form and simply modifies the second, which carries any case or possessive suffixes. So only park gets the accusative.

Could the sentence use parka instead of parkını?

You could say şehir parkına iyileştirme yapıyorum (“I am doing an improvement to the city park”), using the dative parkına (“to the park”). But parkını iyileştiriyorum (“I’m improving the park”) treats the park as a definite object (accusative), which is more direct when you mean “I’m making the city park better.”