Höfuðverkurinn versnar þegar ég sef illa.

Because Icelandic usually puts the definite article on the end of the noun.

• höfuðverkur = headache
• höfuðverkurinn = the headache

So the sentence literally says The headache gets worse when I sleep badly.

If the English meaning you were shown uses my headache, that is coming from context, not from the Icelandic grammar itself. To say my headache explicitly, you would normally say höfuðverkurinn minn.

Yes. Icelandic uses compound words very often.

• höfuð = head
• verkur = pain, ache

So höfuðverkur is literally something like head-pain, meaning headache.

This is very typical Icelandic word-building, and getting used to compounds helps a lot when reading.

Versnar is the present-tense form of the verb versna, which means to get worse or to worsen.

In this sentence:

• höfuðverkurinn is singular
• so the verb is also singular: versnar

A useful way to think of it is:

• Höfuðverkurinn versnar = The headache gets worse

Icelandic often uses a single verb where English might use get + adjective.

Here, þegar means when and introduces a time clause.

So:

• þegar ég sef illa = when I sleep badly

It links the two ideas:

• The headache gets worse
• when I sleep badly

In other contexts, þegar can sometimes have a slightly different time sense, like once or as soon as, but here when is the natural meaning.

Because sofa is the infinitive, meaning to sleep, but after ég you need the finite verb form, not the infinitive.

This verb is irregular in the present tense:

• að sofa = to sleep
• ég sef = I sleep
• þú sefur = you sleep
• hann/hún/það sefur = he/she/it sleeps

So ég sef illa means I sleep badly.

English speakers often expect something closer to ég sofa, but that would be like saying I to sleep.

Because illa is an adverb, and it modifies the verb sef.

• illa = badly / poorly
• sef illa = sleep badly

An adjective would describe a noun, but here we need a word that describes how the sleeping happens.

Compare:

• illur = adjective form, bad / ill
• illa = adverb form, badly

So Icelandic works like English here: you say sleep badly, not sleep bad.

Because after a subordinating word like þegar, Icelandic normally uses regular subordinate-clause word order:

• þegar + subject + verb
• þegar ég sef illa

So the subject ég comes before the verb sef.

This is different from main-clause word order, where Icelandic often follows the verb-second pattern. In the subordinate clause here, you do not invert to þegar sef ég.

Yes. You can also say:

Þegar ég sef illa, versnar höfuðverkurinn.

That means the same thing.

Notice what happens in the main clause after the fronted þegar-clause:

• Þegar ég sef illa, versnar höfuðverkurinn.

The verb versnar comes before the subject höfuðverkurinn. That is a good example of Icelandic verb-second word order in a main clause.

So both are correct:

• Höfuðverkurinn versnar þegar ég sef illa.
• Þegar ég sef illa, versnar höfuðverkurinn.

A few sounds here are especially important for learners:

• þ in þegar is like th in thin
• ð in höfuðverkurinn is like th in this, though often very soft in actual speech
• ll in illa is not pronounced like ordinary English ll; for many learners, a rough approximation like itl-a is closer than plain illa
• ö in höfuð- is like the ö in German, not like English o

You do not need perfect pronunciation immediately, but it helps to know that Icelandic spelling is quite systematic, even when the sounds are unfamiliar.