But you're a native speaker (I presume) and know what the word is supposed to be! :)
Pretend you don't speak Turkish and listen to the recording again. It sounds like the word ends in -ar rather than -er. That might be normal (and probably is), but it doesn't sound exactly like it's spelled to someone who isn't sure how the word is supposed to be spelled.
Selcen, have you ever noticed that sometimes, the "e" in Turkish sounds a little bit like the "ә" in Azerbaijani? I mean that it sometimes sounds more like an /æ/. In English, this sound is similar to the "a" in "cat". But of course, in Turkish, if you write the letter "a", it has a very different sound. I think the person was trying to say it sounds more like gecelær.
There is an interesting Wikipedia page on Turkish phonology (http://en.wikipedia.org/wiki/Turkish_phonology)... The section on vowels shows that the "e" has slightly different qualities...
Having been exposed to native speakers quite a lot now (without really learning much because they were all bilingual) I can tell you that this doesn't go away. The vowels are a bit different, as i guess they always are when you move to a new language, and it takes some time to adjust.
Words ending in -er or -ar, in particular, usually sound funny to my ear. I also usually hear a very faint "h" at the end of such words, but never got a native speaker to agree with me that hat is a real thing...
When you say you hear a faint "h" at the end of words that end in "-er" or "-ar", do you mean it sounds as if the "r" has a fricative quality? If so, it's normal. Look at this page http://en.wikipedia.org/wiki/Turkish_phonology#Consonants
What is literally, word-by-word translation of "görüşürüz"? I suppose it is the verb "görüşmek" (to meet, to see each other) with 1st person plural ending? Without literally translations it is so hard to learn this "phrases" section. I hate to learn by heart things which I don't understand exactly.
Maybe it's just me, but somehow I had problems with understanding the TTS pronunciation of görüşürüz. Here are some samples of how real people pronounce it: http://de.forvo.com/word/g%C3%B6r%C3%BC%C5%9F%C3%BCr%C3%BCz/#tr
If it is common/usual/the only possibility in Turkish to actually wish good nights/days/mornings (the plural form), then I wonder how you emphasize something like "have a few good days (in your holiday, for instance)"? By adding an indefinite quantifyer like in the English example?