Sırtımdaki ağrı yatışınca rahatladım.

Sırtımdaki means the one on/in my back or the one that is on my back, depending on what noun follows it.

It is built like this:

• sırt = back
• sırtım = my back
• sırtımda = on/in my back
• sırtımdaki = the one that is on/in my back

So in sırtımdaki ağrı, the whole phrase means the pain in my back or the pain on my back.

The ending -ki turns a location phrase into something like an adjective:

• masadaki kitap = the book on the table
• cebindeki para = the money in his/her pocket
• sırtımdaki ağrı = the pain in my back

Because the sentence needs a noun phrase meaning the pain in my back.

• sırtımda ağrı means something more like there is pain in my back or pain in my back
• sırtımdaki ağrı means the pain that is in my back

So:

• Sırtımda ağrı var. = I have pain in my back.
• Sırtımdaki ağrı yatışınca... = When the pain in my back subsided...

In your sentence, ağrı is a specific pain already being talked about, so sırtımdaki ağrı is the natural form.

Because sırtımdaki ağrı already clearly refers to the speaker’s pain through sırtım = my back. Adding -ım to ağrı would usually sound unnecessary here.

• sırtımdaki ağrı = the pain in my back
• sırtımdaki ağrım = my pain in my back

The second form is possible in some contexts, but it is more emphatic or more specific than needed. In ordinary Turkish, sırtımdaki ağrı is the most natural choice.

Yatışınca means when it subsided, once it calmed down, or after it eased.

It comes from the verb yatışmak, which means things like:

• to calm down
• to subside
• to die down
• to ease

So here ağrı yatışınca means:

• when the pain subsided
• once the pain eased

It is:

• yatış- = verb stem
• -ınca / -ince = a converb ending meaning when, once, after

So:

• yatışınca = when it subsided / once it calmed down

This ending is very common in Turkish:

• gelince = when he/she came
• bitince = when it finished
• görünce = when I saw / when he saw, depending on context

The subject and tense are usually understood from context and from the main verb.

They are often translated the same way, but Turkish -ınca/-ince is a verb ending, not a separate word.

English:

• When the pain subsided, I felt relieved.

Turkish:

• Sırtımdaki ağrı yatışınca rahatladım.

So instead of adding a separate word like when, Turkish attaches -ınca/-ince directly to the verb.

This form often gives a sense like:

• when
• once
• after

depending on context.

Because Turkish often leaves the subject unspoken when it is clear from context.

Here, the subject of yatışınca is sırtımdaki ağrı:

• Sırtımdaki ağrı yatışınca = when the pain in my back subsided

So the noun ağrı is the thing doing the action yatışmak.

This is why no separate pronoun like it is needed.

Yes, historically and semantically they are related.

• yatmak usually means to lie down, to go to bed, to stay, depending on context.
• yatışmak has developed meanings like to calm down, to subside, to settle.

In many contexts, yatışmak is used for things like:

• pain subsiding
• anger calming down
• a situation settling

So in this sentence, it does not mean the pain literally lay down. It means the pain eased/subsided.

Rahatladım means I felt relieved, I relaxed, or I was relieved.

It comes from:

• rahat = comfortable, relieved, at ease
• rahatlamak = to relax / to feel relieved
• rahatladım = I relaxed / I felt relieved

In this sentence, I felt relieved is the most natural translation.

Both are possible in Turkish in some contexts, but rahatlamak is a very common and natural verb for to feel relieved or to relax.

• rahatladım = I felt relieved / I relaxed
• rahat oldum = I became comfortable / I felt at ease

Here, rahatladım sounds more idiomatic because the sentence is about relief after pain easing.

Because the whole sentence describes a completed past event:

• the pain subsided
• then the speaker felt relieved

So -dım marks simple past first person singular:

• rahatla-dı-m
• relax/feel relieved + past + I

This gives:

• I felt relieved
• I relaxed

The sentence has two parts:

1. Sırtımdaki ağrı yatışınca
   = when the pain in my back subsided
   This is the time clause.

2. rahatladım
   = I felt relieved
   This is the main clause.

So the structure is:

[When X happened], [I felt relieved].

This is a very common Turkish pattern.

Yes, very normal.

Turkish often puts the time/background clause first and the main result second:

• Sırtımdaki ağrı yatışınca rahatladım.

This is natural because it first gives the condition/time:

• when the pain subsided

and then the result:

• I felt relieved

You could rearrange it in speech for emphasis, but this order is the most standard and smooth.

Yes, that is basically the same idea. The exact English wording may vary:

• I felt relieved when the pain in my back subsided.
• I was relieved once the pain in my back eased.
• I relaxed when the pain in my back died down.

The Turkish sentence is flexible enough to match several natural English translations.

In many situations both relate to pain, but they are not always interchangeable.

• ağrı is often used for physical aches and pains, especially more neutral or medical-sounding pain
• acı can mean pain too, but it can also mean emotional pain, suffering, or a sharper kind of pain

For back pain, ağrı is the normal choice:

• sırt ağrısı = back pain
• baş ağrısı = headache

So sırtımdaki ağrı is exactly what you would expect here.