Kışlık çoraplar kalın, bu yüzden ayaklarım üşümüyor.

Kışlık means for winter / winter-type / suitable for winter.

It comes from:

• kış = winter
• -lık / -lik / -luk / -lük = a suffix that can mean for, related to, intended for

So:

• kışlık çoraplar = winter socks
• more literally, socks for winter

This suffix is very common in Turkish:

• yazlık = for summer, summer-type
• günlük = daily / diary, depending on context
• tuzluk = salt shaker / salt container

Here, kışlık functions like an adjective describing çoraplar.

Yes, you could say both, depending on context.

• çorap = sock / socks in a general sense
• çoraplar = socks, specifically plural

In this sentence, kışlık çoraplar clearly means the winter socks or winter socks as a plural set.

Turkish does not always need the plural marker when English does, especially in general statements. But here, using -lar is very natural because the speaker is talking about an actual pair/set of socks.

So:

• Kışlık çorap kalın. sounds less natural here
• Kışlık çoraplar kalın. is the normal choice

In Turkish, the verb to be is often omitted in the present tense.

So:

• Kışlık çoraplar kalın. = The winter socks are thick.

There is no separate present-tense word for are here. Turkish simply puts the noun phrase and adjective together.

Compare:

• Hava soğuk. = The weather is cold.
• Ev büyük. = The house is big.
• Çoraplar kalın. = The socks are thick.

This is a very common Turkish pattern.

Here, kalın means thick.

So:

• kalın çorap = thick socks
• kitap kalın = the book is thick

It can also describe thickness in a more general sense, including physical thickness or even a deep voice in some contexts. But in this sentence, it clearly refers to the socks being thick enough to keep the feet warm.

Bu yüzden means because of this, for this reason, or more naturally in English, so or therefore.

In this sentence:

• Kışlık çoraplar kalın, bu yüzden ayaklarım üşümüyor.
• The winter socks are thick, so my feet aren’t getting cold.

Other similar connectors include:

• bu nedenle = for this reason
• onun için = therefore / so / that’s why

Bu yüzden is very common and natural in everyday Turkish.

Ayaklarım means my feet.

It is built like this:

• ayak = foot
• -lar = plural
• -ım = my

So:

• ayaklarım = my feet

If you said ayağım, that would mean my foot (singular).

Since the speaker is talking about both feet, ayaklarım is the natural choice.

Notice that Turkish often uses a plural noun plus a possessive ending this way:

• ellerim = my hands
• gözlerim = my eyes
• ayaklarım = my feet

This is a regular combination of suffixes:

1. ayak = foot
2. ayaklar = feet
3. ayaklarım = my feet

The possessive suffix for my changes according to vowel harmony:

• -ım / -im / -um / -üm

Because the last vowel before the suffix here is a, the correct form is -ım.

So:

• ayaklarım not
• ayaklarim

The dotless ı is important in Turkish spelling.

Üşümüyor means is not getting cold or is not feeling cold.

It comes from the verb:

• üşümek = to feel cold, to get cold

Then it is formed as:

• üşü- = verb stem
• -müyor = negative present continuous / ongoing present

So:

• üşüyorum = I am cold / I am getting cold
• üşümüyorum = I am not cold / I am not getting cold
• ayaklarım üşümüyor = my feet are not getting cold

In English, we might translate it a few ways depending on context:

• my feet aren’t cold
• my feet aren’t getting cold
• my feet don’t feel cold

Because the verb is üşümek, not üşmek.

The stem is üşü-, so the negative present continuous is built from that stem:

• üşü + m + üyor → üşümüyor

The -m- is the negative marker, and -yor is the present continuous marker. The vowel before -yor changes according to sound patterns, which is why you get -üyor / -ıyor / -iyor / -uyor in practice.

Compare:

• geliyor = is coming
• gelmiyor = is not coming
• oturuyor = is sitting
• oturmuyor = is not sitting
• üşüyor is from a different verb, düşmek-like patterns? So it would not mean the same thing.

So üşümüyor is the correct form from üşümek.

It can be used for both.

For a person:

• Üşüyorum. = I’m cold.

For body parts:

• Ellerim üşüyor. = My hands are cold.
• Ayaklarım üşümüyor. = My feet aren’t cold.

This is very natural in Turkish. English sometimes prefers my feet are cold, but Turkish often uses üşümek in these situations.

The sentence is in a very natural Turkish order:

• Kışlık çoraplar kalın, bu yüzden ayaklarım üşümüyor.

Literally:

• Winter socks thick, because of this my feet aren’t getting cold.

Turkish word order is flexible, but the neutral order is often:

• topic/context first
• important comment later
• verb usually at the end of the clause

So the second clause ends with the verb:

• ayaklarım üşümüyor

You could rearrange parts for emphasis, but this version is smooth and standard.

Yes. Bu yüzden is flexible.

These are all possible, though some are more natural than others depending on emphasis:

• Kışlık çoraplar kalın, bu yüzden ayaklarım üşümüyor.
• Kışlık çoraplar kalın. Bu yüzden ayaklarım üşümüyor.
• Bu yüzden ayaklarım üşümüyor; kışlık çoraplar kalın.
  This changes the focus and sounds less neutral in everyday speech.

The original sentence is a very natural way to express cause and result.

A few useful pronunciation points:

• ı = a dotless vowel, somewhat like a relaxed uh/i sound, but not exactly the English i
• ş = sh
• ç = ch
• ü = like the German ü or French u

Rough guides:

• kışlık ≈ kush-luk (very approximate)
• çoraplar ≈ cho-rap-lar
• üşümüyor ≈ ü-shü-mü-yor

Also note:

• Turkish pronunciation is usually very regular.
• Each written vowel is pronounced.
• Stress patterns can vary, but clear syllable-by-syllable pronunciation will usually be understood well.

Yes. That is a perfectly reasonable translation.

The Turkish sentence expresses a cause-and-result relationship:

• Kışlık çoraplar kalın = the winter socks are thick
• bu yüzden = therefore / so
• ayaklarım üşümüyor = my feet don’t get cold / aren’t cold

Depending on context, good English translations include:

• The winter socks are thick, so my feet aren’t cold.
• The winter socks are thick, so my feet don’t get cold.
• Because the winter socks are thick, my feet don’t get cold.

All of these capture the meaning well.