Babam hap almadan uyuyamadı, annem ise merhem sürmeden rahat etmedi.

It means without doing.

• al-madan = without taking
• sür-meden = without applying / without putting on

This form is made from the verb stem plus -madan/-meden, and it turns the verb into an adverbial phrase.

Because of the whole pattern, not just the suffix by itself.

Literally, the clauses say:

• Babam hap almadan uyuyamadı = My father couldn't sleep without taking a pill
• Annem ise merhem sürmeden rahat etmedi = My mother didn't get relief without applying ointment

In natural English, that often comes out as:

• He couldn't sleep until he took a pill
• She didn't feel better until she applied ointment

So -madan/-meden still means without doing, but when the main verb is also negative, the overall sense is often not ... until ....

Because they mean different things.

• uyumadı = he didn't sleep
• uyuyamadı = he couldn't sleep

So uyuyamadı expresses inability, not just simple negation.

A useful comparison:

• uyuyabildi = he was able to sleep
• uyuyamadı = he wasn't able to sleep / he couldn't sleep

The verb is uyumak and its stem is uyu-. Since that stem ends in a vowel, Turkish inserts a buffer y before certain vowel-initial suffixes.

So you get:

• uyu- + ... → uyu-y-...

That is why you see uyuyamadı, not a form without y.

Because Turkish often shows possession with a suffix on the noun itself.

• baba = father
• babam = my father
• anne = mother
• annem = my mother

The final -m here is the 1st person singular possessive suffix, meaning my.

Because they are being used as indefinite / non-specific objects.

• hap almak = to take a pill / pills
• merhem sürmek = to apply ointment

In Turkish, a bare object often means something indefinite or non-specific. If you meant a specific known pill or a specific ointment, you would more likely see:

• hapı
• merhemi

ise adds contrast. Here it means something like:

• whereas
• as for
• on the other hand

So annem ise means roughly whereas my mother or as for my mother. It helps set up a comparison between the father and the mother.

Because rahat etmek is a common Turkish expression.

Here it means something like:

• to feel comfortable
• to be at ease
• to get relief

So rahat etmedi means she didn’t feel comfortable / she didn’t get relief.

Turkish often uses combinations like noun/adjective + etmek to form verbal expressions.

Often yes, depending on style and nuance.

• rahatlamak = to relax / to feel relieved
• rahat etmemek = not to be comfortable / not to get relief

In this sentence, rahat etmedi is a natural way to say that the mother did not feel okay until using the ointment. Rahatlamadı would also be understandable, but rahat etmedi is a very normal expression.

Because sürmek is the normal Turkish verb for applying something like:

• ointment
• cream
• lotion
• lipstick
• paint

Its core idea is to spread / smear / put on a surface. So merhem sürmek is the standard way to say to apply ointment.

Because Turkish usually puts the main verb at the end of the clause, and phrases like hap almadan or merhem sürmeden normally come before the verb they modify.

So the structure is roughly:

• Subject + without-doing phrase + main verb

That gives:

• Babam + hap almadan + uyuyamadı
• Annem ise + merhem sürmeden + rahat etmedi

This is very normal Turkish word order.

That is due to vowel harmony.

The suffix has two forms:

• -madan
• -meden

Which one appears depends on the vowels in the verb stem.

• al- → almadan
• sür- → sürmeden

Same suffix, different vowel shape to match Turkish vowel harmony.