Ben ekmeğin üstüne tahin sürdüm, annem de biraz pekmez ekledi.

Yes. Turkish often drops subject pronouns because the verb already shows the person.

• sürdüm = I spread
• The -üm ending already tells you the subject is I

So ben is optional here. It is probably included for emphasis or contrast, especially because the second clause mentions annem. It helps create a nice contrast:

• Ben ... annem de ...
• I ... and my mother too ...

Literally, it is something like onto the bread’s top/surface.

It is built like this:

• ekmek = bread
• ekmeğin = of the bread
• üstü = its top / upper side / surface
• üstüne = onto its top / onto its surface

So ekmeğin üstüne is a very natural Turkish way to say on the bread.

Because it takes the genitive ending, which here means of the bread.

• base form: ekmek
• genitive: ekmeğin

Two things happen:

1. The genitive ending is added: -in
2. Final k often changes to ğ before a vowel

So:

• ekmek + in → ekmeğin

This is a very common sound change in Turkish.

Because these are different cases:

• üstüne = onto / on(to)
• üstünde = on it / on top of it

In this sentence, üstüne is used because the tahin is being spread onto the bread’s surface.

Compare:

• Ekmeğin üstüne tahin sürdüm = I spread tahini on the bread
• Tahin ekmeğin üstünde = The tahini is on the bread

So -e / -a here marks the target surface.

Here, sürmek means to spread or to smear/apply onto a surface.

So:

• tahin sürmek = to spread tahini
• ekmeğe tereyağı sürmek = to spread butter on bread

This verb has several meanings in Turkish in other contexts, such as to drive, so learners often notice that. But here the meaning is clearly spread/apply.

Sürdüm is in the simple past tense.

It breaks down as:

• sür- = verb stem
• -dü- = past tense marker
• -m = first person singular

So sürdüm means I spread or I did spread.

The same tense appears in ekledi:

• ekle- = add
• -di = past tense
• third person singular has no extra ending here

So ekledi = he/she added.

Because they are being treated as indefinite or non-specific amounts.

In Turkish, a direct object often stays bare if it is indefinite:

• tahin sürdüm = I spread some tahini
• pekmez ekledi = she added some molasses

If you wanted to talk about a specific tahini or molasses, you would usually use the accusative:

• tahini sürdüm = I spread the tahini / that specific tahini
• pekmezi ekledi = she added the molasses / that specific molasses

This bare-object pattern is very common in Turkish.

Biraz means a little or some.

So:

• biraz pekmez = a little molasses / some molasses

It makes the amount explicitly indefinite and small. That is one more reason the noun stays without the accusative here.

Here de means too, also, or as well.

So:

• annem de = my mother too / my mother also

It adds the idea that after I did something, my mother also did something.

Important: this de is written separately because it is a particle, not a case ending.

• annem de = my mother too
• annemde = on/in/at my mother

Those are completely different.

Because Turkish uses possessive endings.

• anne = mother
• annem = my mother

The -m ending means my.

So Turkish does not need a separate word for my here. The possession is built directly into the noun.

Yes. Turkish word order is flexible, although the given version is very natural.

The most neutral pattern is often close to:

• Ben ekmeğin üstüne tahin sürdüm
• Annem de biraz pekmez ekledi

But Turkish can move parts around for emphasis. For example, if you want to emphasize tahin, ekmeğin üstüne, or annem de, the order can change.

Still, the version you have is a very normal everyday sentence.

Because Turkish can join short related clauses with just a comma, especially in natural narration.

So this works well:

• Ben ekmeğin üstüne tahin sürdüm, annem de biraz pekmez ekledi.

You could also say:

• Ben ekmeğin üstüne tahin sürdüm ve annem de biraz pekmez ekledi.

That is also grammatical, but the version with the comma sounds very natural and smooth.

In ekmeğin, the ğ is usually not pronounced like a strong consonant.

A rough guide is:

• ek-me-in or ek-mey-in, depending on accent and speed

The ğ often just softens the transition or slightly lengthens the vowel. For many learners, the main thing is not to pronounce it as a hard g.

So avoid something like ek-me-gin.