Yeni ayakkabılar yüzünden kızaran topuğuma hemşire merhem sürdü.

The core sentence is:

• Hemşire merhem sürdü = The nurse applied ointment

Everything else adds detail:

• Yeni ayakkabılar yüzünden = because of the new shoes
• kızaran topuğuma = to my heel, which had become red

So the full idea is:

• The nurse applied ointment to my heel, which had become red because of the new shoes.

Turkish usually puts the verb at the end, so sürdü comes last.

Yüzünden means because of, due to, or as a result of.

So:

• yeni ayakkabılar yüzünden = because of the new shoes

A useful note: yüzünden often suggests that the result is unpleasant or problematic. That fits here, because the new shoes caused the heel to become red and irritated.

Kızaran comes from the verb kızarmak, which means to become red, to redden, or sometimes to blush, depending on context.

Here:

• kızaran topuk = heel that became red / reddening heel / reddened heel

The ending -an / -en makes a participle, which often works like that ...s or which ...s in English.

So Turkish says:

• kızaran topuğum
  literally: my heel that became red

Instead of using a separate relative clause like English often does, Turkish places this participle directly before the noun.

Topuğuma breaks down like this:

• topuk = heel
• topuğum = my heel
• topuğuma = to my heel

More specifically:

• topuk
  • -um = topuğum (my heel)
• topuğum
  • -a = topuğuma (to my heel)

Also notice the sound change:

• topuk → topuğ-

The final k often softens to ğ when a vowel-initial suffix is added.

Because the heel is the place/target where the ointment is applied, not the thing being directly acted on in the same way.

With sürmek in this meaning, Turkish commonly uses:

• the thing applied in the accusative or bare form: merhem
• the place/body part in the dative: topuğuma

So:

• merhem sürdü = applied ointment
• topuğuma = to my heel / onto my heel

This is very natural Turkish:

• elime krem sürdü = applied cream to my hand
• yüzüne boya sürdü = put paint on his/her face

So topuğuma is correct because the heel is the destination of the ointment.

• merhem = ointment, salve
• sürmek = to spread, to smear, to apply
• sürdü = applied / spread / smeared (past tense, third person singular)

So merhem sürdü literally means something like:

• she/he spread ointment
• more naturally: she/he applied ointment

In this sentence, the subject is hemşire, so:

• hemşire merhem sürdü = the nurse applied ointment

It is built into topuğuma.

The possessive suffix -um means my:

• topuk = heel
• topuğum = my heel
• topuğuma = to my heel

Turkish often expresses possession with a suffix on the noun instead of a separate word like my.

Because Turkish, like English, often uses the plural when talking about shoes as a pair:

• ayakkabı = shoe
• ayakkabılar = shoes

So:

• yeni ayakkabılar = new shoes

That is perfectly natural here, since a pair of new shoes caused the heel problem.

By itself, hemşire can mean either nurse, a nurse, or the nurse depending on context.

Turkish has no separate words for a/an and the in the same way English does. So context decides.

In this sentence, English might translate it as:

• The nurse applied ointment...
• or A nurse applied ointment...

If the meaning already shown to the learner uses the nurse, that is a perfectly normal translation.

That is a very normal Turkish pattern.

Turkish usually puts:

• modifiers before the noun they describe
• the main verb at the end

So in this sentence:

• yeni ayakkabılar yüzünden modifies the situation
• kızaran comes before topuğuma because it describes the heel
• sürdü comes at the end because it is the main verb

A very literal English-style order would look something like:

• Because of the new shoes, to my reddened heel, the nurse ointment applied

That sounds strange in English, but it is normal Turkish structure.