Dudakların kurusa bile çok su içersen daha iyi hissedersin.

Dudakların means your lips here.

It breaks down like this:

• dudak = lip
• dudaklar = lips
• dudakların = your lips

The ending shows possession, so Turkish does not need a separate word like your unless the speaker wants extra emphasis.
A fuller version would be senin dudakların, but senin is usually omitted.

kurusa bile means even if (they) are dry.

• kuru = dry
• -sa / -se = if
• bile = even

So kurusa bile gives a concessive meaning: even if dry / even if they are dry.

A learner may expect kuruysa bile, which is a more explicit form you will also see. In this sentence, the meaning is still even if your lips are dry.

Because Turkish often leaves out the verb to be in the present.

So:

• Dudakların kuru. = Your lips are dry.

When Turkish adds the conditional idea if, it attaches it to the adjective/predicate:

• kurusa = if [they are] dry

This is very normal in Turkish. English needs are, but Turkish usually does not.

Because Turkish adjectives do not agree with nouns in number.

So:

• kuru dudak = dry lip
• kuru dudaklar = dry lips

The adjective stays the same.
That is why you do not need something like a plural form of kuru.

The same idea applies in the predicate part too: dudakların kuru = your lips are dry.

içersen comes from içmek = to drink.

A simple breakdown is:

• iç- = drink
• -er = aorist/simple present marker
• -se = if
• -n = you

So içersen means if you drink.

In Turkish, this kind of form is very common in conditions:

• gelirsen = if you come
• yaparsan = if you do
• içersen = if you drink

Because in Turkish, the aorist/simple present often expresses a general result or a future result, especially after an if clause.

So:

• çok su içersen daha iyi hissedersin

naturally means:

• if you drink a lot of water, you’ll feel better

Even though the form looks like a present/habitual form, English usually translates it as a future result in this kind of sentence.

daha iyi means better.

Literally:

• daha = more
• iyi = good / well

Turkish usually makes comparatives with daha + adjective/adverb.

So:

• iyi = good / well
• daha iyi = better

This is why Turkish does not need a separate word like English better here.

Because Turkish usually drops subject pronouns when the verb ending already shows the person.

Here, the verbs already tell us the sentence is talking about you:

• içersen = if you drink
• hissedersin = you feel / will feel

So adding sen is unnecessary unless the speaker wants emphasis or contrast.

A learner may know the very common expression kendini iyi hissetmek = to feel good / to feel well.

So you might expect:

• kendini daha iyi hissedersin

That version is also natural.

But Turkish can sometimes omit kendini when the meaning is clear, especially in everyday speech. So daha iyi hissedersin still works and is understood as you feel better.

Because they do two different jobs.

• kurusa bile = even if your lips are dry
  This gives a concessive background: the situation is true, but the next point still stands.

• çok su içersen = if you drink a lot of water
  This gives the real condition for the result.

Then the main result comes at the end:

• daha iyi hissedersin = you’ll feel better

So the structure is basically:

• Even if X, if Y, then Z.

Yes. It is a natural Turkish order.

Turkish often puts background information and subordinate clauses before the main result, and the main verb usually comes near the end.

So this pattern is very typical:

• [even if clause] + [if clause] + [main result]

Here:

• Dudakların kurusa bile = even if your lips are dry
• çok su içersen = if you drink a lot of water
• daha iyi hissedersin = you’ll feel better

You can change the order for emphasis in Turkish, but this version is straightforward and natural.