As a native Azeri we have a lot of sounds like "a" in HAT but in turkish it seems to me the letter "e" is sometimes pronuced as "A" in HAT and not "e" in PET.

Can anybody make this clear for me?

In Turkish, the vowel used in the word "hat" does not exist. Nevertheless, there are people, especially those with Arabic or Kurdish lineage, speaking Turkish as a secondary language. Those people usually substitute the vowel "a" with the Arabic or Kurdish wovel "ae".

There are 8 wovels in the Turkish language and their use is bound by the two wovel harmony rules which govern the use of what we may call as hard and soft wovels as well as round and plain vowels. Accordingly, a hard vowel (a,ý,o,u) cannot coexist with a soft wovel (e,i,ö,ü in the same word. Furthermore, plain vowels (a,i,ý,e) cannot be used together with round vowels (o,ö,u,ü. You can easily determine the originality of a word by checking it against these two vowel harmony rules. 

In Turkish, the wovel used in the word "hat" does not exist. Nevertheless, there are people, especially those with Arabic or Kurdish lineage, speaking Turkish as a secondary language. Those people usually substitute the wovel "a" with the Arabic or Kurdish wovel "ae".

This statement is not correct. Sometimes the pronunciation of the letter "a" is very close to the pronunciation of "a" in "hAt" (British pronunciation of this word not American)

for example  bEn (sounds more like ban)

There are a lot of rules about this. If you want to perfect your Turkish pronunciatin, pm me

TT

You certainly speak another version of Turkish... I never say "baen" or "saen". Nevertheless, the "e" used in Turkish is not the same as the English "e". To my knowledge, there is a certain "gay talk"  which probably mimics the way a coquette renders certain words.  You would hear those baens and saens (ben and sen) only in those circles (in addition to those under Arabic influence.)

By the way, I made a spelling mistake and wrote "wovel" instead of "vowel".

usually some girls speak like that

As a matter of fact, one needs to have certain criteria when comparing vowels. One of the important parameters is the length of a vowel. In Turkish all vowels are short ones. In English there are both short and long vowels. The ´ae´ sound of English is a long vowel whereas all the variations of ´e´ in Turkish are short.

The ´ae´ sound of English is a long vowel whereas all the variations of ´e´ in Turkish are short.

this statement is absolutely wrong. ae is a short vowel in British english. not all the variations of "e" are short. example   "memur"  : you have to elongate the "e" in this word we borrowed from Arabic. There are many words like this in Turkish

Have you ever  spoken to an American speaker of English? Or are you giving this information based on received pronunciation? I can´t imagine anyone saying the words bad or sad using a short vowel.You would say, vowels succeeded by fortis plosives are rendered a tad shorter while those used before lenus ones are rendered slightly longer. This is all linguistic bullshit. Real people don´t talk like that unless they want to mimic the voices on language teaching tapes.

Your idea about Turks rendering the word "ben" like  "hat" is pointless...

I have spoken to thousands of American and British speakers. memur is pronounced like "meeemur"

not memur . Not all the British elongate the vowel in "hAt" either. (the pronunciation of this word is different in American English.

ben > (more like pan)

beni > (as in pen)

"memur" is not a Turkish word, full stop.