Misafirlik bitince eniştemi kapıda uğurladık, ama teyzem biraz daha oyalandı.

Here, misafirlik does not mean hospitality in the abstract sense. In everyday Turkish, it often means a visit, visiting someone, or the state of being a guest.

So Misafirlik bitince means something like:

• when the visit was over
• once the guest visit ended

It refers to the social visit itself.

Bitince comes from the verb bitmek = to end / to be finished.

Breakdown:

• bit- = end, finish
• -ince = when / once / after

So:

• bitince = when it ended / once it was over

This -ince form is very common in Turkish for time clauses:

• gelince = when he/she comes
• gidince = when he/she goes
• yemek bitince = when the meal is over

In this sentence, Misafirlik bitince sets the time for what happened next.

These are specific family terms.

• teyze = maternal aunt (your mother’s sister)
• enişte = usually your sister’s husband or your aunt’s husband

In this sentence, since teyzem appears, eniştem most naturally means my aunt’s husband.

This is one thing English learners often notice: Turkish family vocabulary is usually more specific than English. English aunt can mean several different relationships, but Turkish distinguishes them.

Turkish very often adds possessive endings to family words.

• teyzem = my aunt
• eniştem = my brother-in-law / my uncle by marriage

The -m here means my.

So:

• teyze = aunt

• teyzem = my aunt

• enişte = brother-in-law / uncle by marriage
• eniştem = my brother-in-law / my uncle by marriage

Then eniştemi has one more ending:

• enişte-m-i
• enişte = brother-in-law / uncle by marriage
• -m = my
• -i = accusative marker

So eniştemi means my brother-in-law / uncle by marriage as a specific direct object.

Because eniştemi is the direct object of the verb uğurladık, while teyzem is the subject of oyalandı.

• eniştemi kapıda uğurladık = we saw off my brother-in-law
• teyzem biraz daha oyalandı = my aunt lingered a bit longer

The ending -i here is the definite accusative. Turkish commonly marks a specific direct object with this ending.

So:

• eniştemi = a specific person being seen off
• teyzem = subject, so no accusative

Kapıda means at the door or by the door.

Breakdown:

• kapı = door
• -da = in / at / on

So:

• kapıda = at the door

In this sentence, it suggests they accompanied him to the door and said goodbye there.

Not exactly. Uğurlamak usually means to see someone off, to send someone off, or to accompany someone as they leave.

So uğurladık is stronger and more specific than just we said goodbye.

Breakdown:

• uğurla- = see off / send off
• -dık = we in the simple past

So:

• uğurladık = we saw (him) off

Because the sentence also says kapıda, the image is very natural: they walked him to the door and saw him off there.

Biraz daha literally means a little more, but in context it often means a little longer.

So:

• biraz daha oyalandı = she lingered a little longer
• or she stayed a bit longer

This is a very common expression:

• biraz daha bekle = wait a little longer
• biraz daha ye = eat a little more

The exact English translation depends on context.

Oyalandı comes from oyalanmak.

Common meanings of oyalanmak include:

• to linger
• to dawdle
• to stay occupied
• to be delayed
• to hang around a bit

In this sentence, the most natural meaning is:

• she lingered
• she stayed a bit longer
• she didn’t leave right away

Breakdown:

• oyalan- = linger / stay occupied / dawdle
• -dı = simple past, he/she did

So oyalandı = she lingered.

Ama means but.

It introduces a contrast:

• We saw off my brother-in-law at the door
• but my aunt lingered a little longer

So the sentence contrasts two actions:

1. one person left and was seen off
2. the other person did not leave immediately

Yes, this word order is very natural.

The sentence is organized like this:

• Misafirlik bitince = time setting
• eniştemi = object
• kapıda = location
• uğurladık = verb
• ama = contrast
• teyzem = subject
• biraz daha = adverb
• oyalandı = verb

A very common Turkish pattern is:

time expression + object/location + verb

And Turkish often puts the main verb at the end of the clause.

Could it be rearranged? Yes, to some extent. Turkish word order is flexible. But the original version sounds natural and smooth, especially because it first sets the scene with Misafirlik bitince and then gives the two contrasting actions.