Do European and American restaurants throw away animals organs that they don’t use ?

Most American restaurants do not include the sort of butchering you ask about. They purchase the meats they will use already butchered and often already cut up. If they purchase organ meats, it is to resell them at table.

I don't think restaurants are aIIowed to butcher their own meat. It's against food safety regulations in most if not all states. They order steak tips, chicken breasts, filet mignons, or whatever cuts of meat they want from a farm/slaughterhouse. Pet food companies buy the leftover organs. Some have suggested using the leftovers to make fuel, but there are some issues with that. 

Restaurants don't butcher their own meat.  Typically, organ meats are used to make pet food.

Those types of parts don't arrive at restaurants the slaughter houses dispose of those, I think most of animal left overs are used to make cat/dog food. 

Can't speak for Europe, but the vast majority of restaurants order their meat from distributors, who get their meat from the slaughterhouses. The slaughterhouses sell organ meats to distributors and if a restaurant wants some, they can get some. So most restaurants never see the usable entrails because it's not something their customers would eat.

My grandmother owned the neighborhood diner. Any "can't serve to the customers" stuff like cabbage hearts, potato peels, chicken parts, beef bones and trimmings, and so on went into the stock pot. The liquid in the stock pot was used as a base for soups and stews. 

Chicken livers, hearts, gizzards were not on the menu, but she would cook them up and present them as gifts to customers she knew were fond of them. 

The animal organs are kept/disposed of by the butcher.

My local butcher gives some of the organs to the Science Department of the local secondary school to use for disection in biology lessons.

No they use it in stock

Most restaurants aren't getting whole animals. They are getting animal parts usually without the guts. 

The vast majority of US restaurants order their meats in rather than processing the animals on site, with the one exception obviously being fish.  As restaurants are always looking to save, excess from fish can be used for other recipes or stocks.
 
A few restaurants do raise and butcher their own animals, but they do it off site.  In such cases, just as with fish, every part of the animal is vital.  What little waste there is can in turn be processed into animal feed.

Meat comes to restaurants cut not as a whole animal. 
The processing plant deals with organ meat as well as other parts that can not be used. 
Some parts (in the USA) are not legal to sell but if there are parts that you specifically want you might be able to order them either from a butcher or if there are processing plants in your area, or farmers that sell to the public you might be able to order parts when they next bring in an animal.