Bacağım ağrıyor.

It is:

• bacak = leg
• -ım = my

So:

• bacak + ım → bacağım = my leg

The final form changes a little because of a sound change in Turkish, which is why you do not see bacakım.

This is a very common Turkish sound change.

When some nouns ending in -k take a suffix that begins with a vowel, the k often softens to ğ.

So:

• bacak
  • -ım → bacağım

This kind of change is normal in Turkish and happens in many words. It is not a special irregularity of this sentence.

-ım is the 1st person singular possessive suffix, meaning my.

In Turkish, possession is often shown by a suffix attached directly to the noun.

So:

• bacağım = my leg

Because of vowel harmony, this suffix can appear in different forms:

• -ım
• -im
• -um
• -üm

Here it is -ım because that matches the vowels in bacak.

Turkish often leaves out the separate possessive word when the meaning is already clear from the suffix.

So:

• Bacağım ağrıyor. = My leg hurts.
• Benim bacağım ağrıyor. = also possible, but more emphatic

Using benim can add contrast or emphasis, like:

• Benim bacağım ağrıyor, onun kolu değil.
  My leg hurts, not his arm.

In a normal sentence, Bacağım ağrıyor is perfectly natural.

Because the grammatical subject is bacağım (my leg), not I.

In Turkish, the thing that hurts is the subject:

• Bacağım ağrıyor. = My leg hurts.

So the verb agrees with bacağım, which is 3rd person singular.

This is different from how English learners sometimes first imagine it. Turkish is not saying I am hurting my leg or I hurt. It is saying that the leg hurts.

Ağrıyor is the present continuous form of the verb ağrımak (to ache / to hurt).

Structure:

• ağrı- / ağrı- related verb stem
• -yor = present continuous ending

So ağrıyor literally corresponds to something like:

• is hurting
• is aching

But in natural English, it is often translated simply as:

• hurts
• aches

So Bacağım ağrıyor can match either:

• My leg hurts.
• My leg is hurting.

depending on context.

Turkish uses the -yor form very often for something happening or being felt right now, including physical states and sensations.

So for a current pain, ağrıyor is the normal choice.

English often uses the simple present in this kind of situation:

• My leg hurts.

But Turkish prefers the present continuous:

• Bacağım ağrıyor.

So this is a good example where Turkish and English use different tense/aspect choices even though the meaning is straightforward.

A learner usually notices two tricky letters here: ğ and ı.

• ğ (yumuşak ge) is usually very soft; it often lengthens the vowel before it rather than sounding like a strong consonant.
• ı is the dotless i, a vowel that does not really have an exact English equivalent.

A rough learner-friendly pronunciation is something like:

• ba-JAA-uhm

But that is only approximate.

A few points:

• ca in baca- sounds like ja in jar
• ğ is very soft
• final -ım has the dotless ı, not English ee

The difficult parts are again ğ and ı.

A rough approximation is:

• AA-ruh-yor or AA-ruh-yor

But not exactly.

Helpful notes:

• ağ is often like a lengthened a
• r in Turkish is tapped/lightly rolled
• ı is the dotless vowel
• yor sounds roughly like yor in your, but not exactly

So the whole sentence flows roughly like:

• ba-JAA-uhm AA-ruh-yor

Again, that is only an approximation for English speakers.

Yes. This is a very common pattern:

• [body part + possessive] + ağrıyor

Examples:

• Başım ağrıyor. = My head hurts.
• Kolum ağrıyor. = My arm hurts.
• Ayağım ağrıyor. = My foot hurts.
• Dişim ağrıyor. = My tooth hurts.

So this sentence is a very useful model.

Bacağım ağrıyor is the most neutral and natural order.

You may also hear:

• Benim bacağım ağrıyor.

if the speaker wants emphasis on my.

Turkish word order is flexible, but not every variation sounds equally neutral in everyday speech. For learners, the safest pattern is:

• Body part + verb

So:

• Bacağım ağrıyor.

is the best standard form to learn first.