Höfuðverkurinn minnkar þegar ég sef betur.

Because Icelandic very often makes compound nouns by joining smaller words together.

Here:

• höfuð = head
• verkur = pain, ache

So:

• höfuðverkur = headache

Then Icelandic adds the definite article to the end:

• höfuðverkur = a headache / headache
• höfuðverkurinn = the headache

This kind of long compound is extremely normal in Icelandic.

The ending -inn is the definite article, meaning the.

So:

• höfuðverkur = headache / a headache
• höfuðverkurinn = the headache

Unlike English, Icelandic usually puts the at the end of the noun instead of using a separate word before it.

Minnkar is the present tense of the verb minnka, which means to decrease, lessen, get smaller, improve/reduce depending on context.

So:

• að minnka = to decrease
• minnkar = decreases / gets less

In this sentence, höfuðverkurinn minnkar means something like:

• the headache decreases
• the headache gets better
• the headache lessens

All of those are natural English ways to understand it.

Because Icelandic often uses the present tense for a general truth, habitual situation, or something that happens repeatedly.

So:

• Höfuðverkurinn minnkar þegar ég sef betur.

means roughly:

• The headache gets better when I sleep better
• My headache improves when I sleep better

This is not necessarily about one single moment right now. It can describe a general pattern.

Because after ég (I), the verb must be in the correct 1st person singular present form.

The verb is að sofa = to sleep.

Its present tense forms include:

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

So:

• ég sef is correct
• ég sofa is not correct

This is just an irregular verb pattern you need to learn.

Because this sentence uses the indicative, not the subjunctive.

• ég sef = indicative, used for normal factual statements
• ég sofi = subjunctive, used in certain special contexts

Here the speaker is stating a real pattern or fact:

• when I sleep better, the headache gets better

So the normal form is:

• þegar ég sef betur

not sofi.

Betur means better.

It is the comparative adverb of vel (well).

So:

• vel = well
• betur = better

Here it modifies the verb sef:

• ég sef betur = I sleep better

Because betur is an adverb, while betra is usually a form of the adjective betri (better).

In this sentence, better describes how I sleep, so you need an adverb:

• ég sef betur = I sleep better

Compare:

• betri svefn = better sleep
  here betri describes a noun, so it is an adjective
• ég sef betur = I sleep better
  here betur describes a verb, so it is an adverb

This is similar to English good/well vs better, though Icelandic uses its own forms.

Here þegar means when.

It introduces a time clause:

• þegar ég sef betur = when I sleep better

In other contexts, þegar can also mean already, depending on position and usage, but in this sentence it clearly means when.

Yes, but it is actually the normal order for a subordinate clause.

In the clause:

• þegar ég sef betur

the order is:

• þegar = when
• ég = I
• sef = sleep
• betur = better

So the subject ég comes before the verb sef.

That is different from the verb-second pattern you often see in Icelandic main clauses. Since this is a subordinate clause introduced by þegar, the word order is more straightforward.

Yes. Icelandic can also put the when-clause first.

For example:

• Þegar ég sef betur, minnkar höfuðverkurinn.

That means the same thing.

When the subordinate clause comes first, the main clause still follows Icelandic word-order rules, so the verb minnkar comes before the subject höfuðverkurinn.

Literally, it means the headache.

But in natural use, Icelandic often uses the definite noun where English might prefer my if the context already makes possession obvious.

So depending on context, it may be understood as:

• the headache
• my headache

If the speaker wanted to be extra explicit, they could say:

• höfuðverkurinn minn = my headache

But the shorter form can still sound natural if the meaning is obvious.

A rough learner-friendly guide is:

• HÖ-fuð-ver-kur-inn

A few useful pronunciation notes:

• ö sounds roughly like the vowel in French peur or German schön; there is no exact English equivalent.
• ð is like the th in this, though in some positions it can be weaker.
• Stress in Icelandic usually falls on the first syllable, so HÖ- is stressed.

You do not need perfect pronunciation right away, but it helps to recognize the parts:

• höfuð
  • verkur
    • -inn

Yes, it is a natural and common way to say that pain or a symptom decreases or lessens.

For example:

• Verkurinn minnkar. = The pain is decreasing / The pain gets less.
• Bólgan minnkar. = The swelling goes down.

So in this sentence, minnkar is a very natural choice.

Because Icelandic usually attaches the definite article to the noun itself.

English:

• the headache

Icelandic:

• höfuðverkurinn

This is one of the biggest differences English speakers notice early on. Instead of a separate article before the noun, Icelandic often uses a suffix article.

Ég is in the nominative case, because it is the subject of the verb sef.

In:

• þegar ég sef betur

ég is the person doing the sleeping, so nominative is exactly what you would expect.

Yes, in practice betur is the normal comparative form for vel (well) and is usually translated simply as better.

So:

• ég sef vel = I sleep well
• ég sef betur = I sleep better

English does not normally say more well, and neither would Icelandic use a separate equivalent here. Betur is the standard comparative form.