Learn Turkish: Kolundaki yanık iyileşiyordu ama dizi hâlâ morarıyordu. - The burn on his arm was healing, but his knee was still bruising.

The -ki suffix attaches to the locative case (kolunda, meaning "on his arm") to turn it into an adjective describing the burn. So, kolundaki yanık literally means "the on-his-arm burn" or "the burn that is on his arm."

I recognize the word iyi (good) in iyileşiyordu. How does the suffix -leş work here?

The -laş / -leş suffix is added to nouns or adjectives to mean "to become" or "to turn into" that state. Here, iyi (good) + leş makes iyileşmek, which means "to get better" or "to heal."

Is morarıyordu related to the color mor (purple)?

Yes! In Turkish, you can make verbs out of colors or certain adjectives by adding the -ar or -er suffix, meaning "to turn [that color]." Mor (purple) becomes morarmak (to turn purple, or to bruise). Another common example is sarı (yellow) becoming sararmak (to turn yellow or fade).

Doesn't dizi mean "TV series"? Why is it translated as "knee"?

This is a tricky coincidence! While dizi is the standalone noun for a TV series, here it is the root word diz (knee) plus the third-person possessive suffix -i (his/her knee). The surrounding context of physical injuries (yanık, iyileşmek, morarmak) makes it clear that we are talking about a body part.