Gözlüğümü takmadan kitap okuyamıyorum.

It breaks down like this:

• gözlük = glasses / eyeglasses
• -üm = my
• -ü = the accusative ending, used for a definite direct object

So gözlüğümü means my glasses as the object of the verb.

Turkish often puts ideas like my and object marking into suffixes instead of separate words.

Because Turkish normally uses gözlük as a singular noun for a pair of glasses / eyeglasses.

So:

• gözlük = glasses / a pair of glasses
• gözlükler = multiple pairs of glasses

This is just a difference between English and Turkish. English uses a plural-looking word here, but Turkish usually does not.

Because the noun is the direct object of takmadan.

• gözlüğüm = my glasses
• gözlüğümü = my glasses as a specific/definite object

In Turkish, definite direct objects usually take the accusative ending.

This is a common sound change in Turkish.

When some words ending in -k take a vowel-initial suffix, the k often softens to ğ:

• gözlük
• gözlüğüm
• gözlüğümü

So this is normal spelling and pronunciation behavior, not a different word.

Takmadan means without putting on or without wearing.

It comes from:

• takmak = to put on / wear
• -madan / -meden = without doing

So:

• takmadan = without putting on

In the sentence, gözlüğümü takmadan means without putting on my glasses.

Because takmak is the normal Turkish verb for things you put on / attach to yourself, including:

• gözlük takmak = wear glasses
• yüzük takmak = wear a ring
• küpe takmak = wear earrings

So even though English uses wear, Turkish often uses takmak for these items.

It comes from oku- (read) plus a negative ability pattern.

A simple breakdown is:

• oku- = read
• -ya- / -a-
• -ma- = negative
• -yor = present/progressive
• -um = I

So okuyamıyorum means I can't read / I am unable to read.

A very important pattern in Turkish is:

• okuyabiliyorum = I can read
• okuyamıyorum = I can't read

Because Turkish usually does not need a subject pronoun if the verb already shows the person.

Here, -um at the end of okuyamıyorum already tells you the subject is I.

So:

• okuyamıyorum = I can't read
• ben okuyamıyorum = I can't read with extra emphasis on I

Because kitap here is a bare object, which often gives a more general or indefinite meaning.

• kitap okuyamıyorum = I can't read books / I can't read a book in a general sense
• kitabı okuyamıyorum = I can't read the book / that specific book

So the bare kitap sounds more like the general activity of reading.

A rough guide:

• ö = like German ö or French eu
• ü = like German ü or French u
• ğ usually is not pronounced like a hard g

In gözlüğümü, the ğ mostly smooths or lengthens the vowel sound, so it sounds roughly like:

• göz-lüü-mü

The most important thing is: do not pronounce ğ like a strong English g.

Yes. Turkish word order is flexible, although this version is very natural:

• Gözlüğümü takmadan kitap okuyamıyorum.

The main verb usually comes at the end, but other parts can move for emphasis.

This sentence presents the without my glasses part first, which sounds natural and clear: first the condition, then the result.