Dedem eski mahalleyi anlatırken hep aynı sokağı özlediğini söylüyor.

Dedem means my grandfather.

It is made of:

• dede = grandfather
• -m = my

Turkish very often uses a possessive suffix instead of a separate word like my:

• annem = my mother
• babam = my father
• dedem = my grandfather

So you do not need a separate word for my here.

Those endings mark the definite direct object in Turkish, often called the accusative.

Here:

• eski mahalleyi anlatmak = to describe the / that old neighborhood
• aynı sokağı özlemek = to miss the same street

The objects are specific, not generic, so they take the accusative.

Because of two things:

1. It takes the accusative suffix:
   sokak + ı

2. The final k often softens to ğ before a vowel-initial suffix:
   sokak → sokağı

This kind of change is very common:

• çocuk → çocuğu
• bacak → bacağı
• sokak → sokağı

So sokağı is just sokak with the accusative ending and normal sound change.

Anlatırken means while telling / while describing.

It comes from anlatmak = to tell, to describe plus -ken, which means while / when.

So:

• anlatırken = while he/she is telling or describing

In this sentence, eski mahalleyi anlatırken means while describing the old neighborhood.

They are related, but they do different jobs.

• anlatmak = to tell about, describe, relate in detail
• söylemek = to say, state

So the sentence is not repeating the same idea. It means something like:

• while he is describing the old neighborhood
• he says that he misses the same street

So one verb gives the situation/background, and the other gives the main statement.

Yes — this is a very typical Turkish embedded clause form.

A useful breakdown is:

• özle- = miss, long for
• -dik / -diğ- = a suffix that turns the verb into a noun-like clause
• a person marker
• -ni = accusative, because the whole clause is the object of söylüyor

So özlediğini söylüyor means he says that he misses ...

You do not need to translate özlediğini word-for-word every time. It is often best to learn the whole pattern:

• [verb + -DIK + personal ending + accusative] + söylemek

This is one of the standard ways Turkish says that ... after verbs like say, know, see, hear, etc.

Because Turkish often does not use a separate word like English that in sentences like this.

Instead, it builds the that-clause directly into the verb form:

• özlediğini = that he misses

So:

• aynı sokağı özlediğini söylüyor
  = he says that he misses the same street

The meaning of that is already contained inside özlediğini.

From the context and structure, the natural reading is that dedem is also the subject of özlemek.

So the sentence is understood as:

• My grandfather says that he misses the same street

If Turkish wanted to clearly name a different person inside the embedded clause, it would usually say so explicitly, often with a genitive subject, for example:

• Annesinin aynı sokağı özlediğini söylüyor
  = He says that his mother misses the same street

So in your sentence, with no other subject named, dedem is the default interpretation.

• hep = always
• aynı = same

Together, hep aynı sokağı means always the same street.

The idea is: whenever he talks about the old neighborhood, the street he says he misses is always that same one.

So hep adds a sense of repetition, and aynı shows it is not a different street each time.

Because Turkish -yor does more work than English is ...-ing.

Söylüyor literally looks like is saying, but in real Turkish it is also very common for:

• current situations
• ongoing habits
• repeated behavior in context

So with hep, söylüyor can naturally mean:

• keeps saying
• says
• is always saying

All of those are close in sense, and English chooses whichever sounds most natural.

Turkish usually puts the verb at the end of its clause.

The structure here is roughly:

• Dedem = my grandfather
• eski mahalleyi anlatırken = while describing the old neighborhood
• hep aynı sokağı özlediğini = that he misses the same street
• söylüyor = says

So the main verb of the whole sentence, söylüyor, comes last.

A very literal English order would sound unnatural, something like:

• My grandfather, while describing the old neighborhood, that he misses the same street, says.

English rearranges this, but Turkish likes to place the important verb at the end.

Good question. Eski mahalleyi literally means the old neighborhood, not explicitly his old neighborhood.

Turkish often leaves possession unstated when it is obvious from context. Since the sentence is about dedem remembering and speaking, listeners can easily understand that it is probably his old neighborhood.

If you wanted to make it explicit, you could say something like:

• eski mahallesini

But the original sentence does not have to say that. Turkish often prefers the shorter, context-based version.