I know English is full of exceptions but I always was taught that Turkish is a consistent language.

Then I discovered this:

http://www.turkishtools.tourkika.com/possession/

Are there more examples of irregular Possessive forms. I came across 

Some words [especially loanwords ] lose their final vowel (apocopate) when a suffix which itself begins with a vowel is added to the noun.

Example:

izin - leave, time off - becomes izn-im [NOT izin-im]- my leave 

in the example, the final vowel of the noun root is dropped when adding a suffix which begins in a vowel.

-fikir

fikrimiz - our idea

- keyif
keyfi - his/her joy

oğul
oğlum - my son

boyun
boynu - his neck

nakil
nakli - his transport

ahit
ahdi - his promise 

* In the last two examples above that the final consonant -t has changed into its soft form -d.

kayıp  - loss

kaybı - his loss -

* in the example above, -p  softens into  -b when it takes possesive suffix.

Below, some more examples :

ağız / ağzı

alın / alnı

bağır / bağrım 

beniz / benzi

beyin / beynimiz

boyun / boynu 

böğür / böğrüm 

burun / burnu 

geniz / genzi

göğüs / göğsün

gönül / gönlünüz

karın / karnı 

oğul / oğlu

I guess that comes from some form of euphonic contraction, that took over to the point it´s not correct anymore not to do it, similarly to what happens in French and Italian with the definite article: la+vowel becomes l´, or in English with a/an.

Is there any kind of pattern to know which words drop their vowel, or is it something we have to learn along with the words? I notice all your examples are 2-syllable words, and the last vowel is i/ı/ü/u.

  You have to learn those words, however there are some indications that gives us a clue about this kind of vowel dropping ;

* Some organ names with two syllables, when they take suffix starting with a vowel, their middle [second] sound drops.

burun ---> burun + um  ----> burnum

boyun ---> boyun + u   ----> boynu

ağız   ---> ağız + ımız  ----> ağzımız

alın    ---> alın + ın -----> alnın

karın  ---> karın + ım ----> karnım

beyin ---> beyin + i   ----> beyni

* When some two syllable words having wide vowels [a,e ,o ,ö] in their first syllable and narrow vowels  [ı ,i, u ,ü ] in their second syllable , take suffix starting with a vowel, the middle sound drops.

alın    ---> alın  + ı  ----> alnı

resim ----> resim + im ---> resmim

bağır  ---> bağır +  ım ---> bağrım

gönül ----> gönül  + üm ---> gönlüm

So it should NOT be called "possessive exceptions". It can be called a phonetic development.

The possesive exceptions I know of are:

su -> su-yum, su-yun, su-yu, etc.

ne -> ne-yim, ne-yin, ne-yi, etc

şu -> şu-yum, şu-yun, şu-yu, etc.

bu -> bu-yum, bu-yun, bu-yu, etc.

...

Normally possessive suffixes are of dropping kind (after a vowel -m, -n, -si, -miz, -niz, -leri).

pusu -> pusu-m, pusu-n, pusu-su, etc.

anne -> anne-m, anne-n, anne-si, etc.

komşu -> komşu-m, komşu-n, komşu-su, etc.

tabu -> tabu-m, tabu-n, tabu-su, etc.

...

If this were English, everyone would be calling these excpetions.

If these were Polish, they wouldn´t be unusual enough to call them exceptions

OK No objections to your calling it exceptions but possessive exceptions.

I don´t see the possessive suffixes functioning irregularly with those you call exceptions. (I have provided some examples of possessive suffixes functioning irregularly.)

There are some irregularities with those words that happen when any (not just possessive) suffix is added to them.

Does ömür behave differently when other suffixes are added? As I don´t know.

It is " Orta Hece Ünlüsü Düşümü " middle syllable vowel dropping.  Whether you call it as an exception or a fonetic development, does not change the fact that  there is " vowel dropping"  happening with some words or .  It is not seen only with possessives, it can be seen with other suffixes. For instance , take one of those word:

İzin  + e  ---> izne   [ to day [s] off  from work ]
Yakında izne çıkacağım  ---> I will have days off from work soon. [ in other words, I will have get holiday from work]

Middle syllable dropping, because, when it takes a suffix, it becomes the middle syllable.

izin

İ + zi + ne    ----> izne

=================================

ömür
ö + mü + re  ---> ömre

Seninle bir dakika ömre bedel ! ---> Being with you one minute , worth a life !

* However, when the suffix starts with a consonant, then the vowel dropping don´t occur.

Ömür + den  ---> Ömürden
Ömür  + de ---> Ömürde
Ömür + süz   ---> Ömürsüz
Ömür + lü ---> Ömürlü
Ömür + se  ---> Ömürse