Merhaba!  

I´m quite confused about the sound of "r" when it´s the last letter in a word.  For example, 

bir,  from video, it sounds like "beesh" for me.  How to speak the tailing "r´?   Thanks.

I was confused about this too, but the best way it was explained to me is to think of it as not as a "shh" sound (which is ş but more that it is the trailing airflow from the mouth after pronouncing the ´r´ letter.

I often hear it when the final word of the sentence ends in an ´r´, but if a word that ends with ´r´ is followed by more words, then I don´t usually hear that sound.

   In which video it says as "beesh " ? It is not pronounced as "beesh".  

 It is just "bir"  and it sounds like when you say " beer" in english BUT with short "e". [reading "e" short]

You shoud be able to hear that sound , as all letters are supposed to be read in Turkish, unless the person has disability of not being able to say "r" letter. Since some people has that kind of problem. [ Aziz Yıldırım, Beyaz Öztürk ... ] 

Kar 

Sar

Dar 

---------

Kara 

Sara

Dara

-------

* Only sometimes we drop "r" in the end of "bir" in collaquial language, which doesnt mean we should not read that sound.

Bi dakka ==> its actually "Bir dakka"

Bi saniye  ==> " Bi saniye "

Bi gel  --> " Just come " ===> " Bir gel "

Oh, I mean "r" sounds like "ş" (in the previous message, I use "sh" to present "ş".  In the video:

https://www.youtube.com/watch?v=IZYsNXTloFg&index=9&list=PLDOlE-jNL3MoQ2D1a4Q7swqs-ZfbZaTF1

The phrase "Ne Kadar", the ending "r" sounds like "ş".

definitely not same sound as "ş" - think of it like you are about to roll your ´r´s (like the italians do)... but stop short of actually doing it

I watched the video, "r" sounds normal. To me it didnt sound as "ş".

My advice is , you should watch other videos which has words with "ş" sounds in it, so that you may hear the difference. 

just to add some perspective.  To some (not all) english native speakers.. we are not used to hearing this flow of air after Turkish people pronouncing their ´r´. 

In (British) English this sound doesn´t exist since we tend NOT to sound our ´r´ much. example "car" in England we say "caahh"

I totally know where you are coming from. I believe this could be a variation in Turkish accents. I was listening to something from Sila recently and I thought she was singing a word ending with ş but I looked it up and it was a word ending in ´r´.

My suggestion for pronouncing the sound is you could add a little air after the ´r´ but if you leave it to how you pronounce it as you would in English, you will be fine. 

It is an urban myth. Just a sound illusion