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