Why doesn't flightradar show many flights going over China?
From the Flighradar How it works page:


  Flightradar24 relies on volunteers around the world for the majority
  of our coverage


Note it implies:


There are people willing to contribute
These people can afford to buy receptors, gather the data and upload it to Flightradar


About point 1. In case of China how much people are willing to contribute? How many are affraid to do it to a foreign organization and how much are affraid of gettign the wrong attention from government?

About point 2. Most people on China cannot afford to buy such equipment (compared to USA).

It's easy to check point 2 by comparing macro economic statistics (So I doubt I need to link it here).

Point 1 is very speculative so each one should try to figure that by him/herself.

In fact I always wondered Fligth radar will got a poor coverage in third world countries (point 1) while getting near 100% coverage in USA.

Note also the size of aviation community (pilots, enthusiasts, aviation industry workers, etc) in USA is so much bigger tha in other corners of the world.

Edit's in response to Jaydeep comments

Flighradar giving away equipment like that is great, I think Wow!
Note also the note on the same page:


  Exceptions to these requirements can be made to applications from
  unique locations.


So that begs the question if they are actually granting such gifts to China. For the sake of the completion I will ask them.

EDIT     and The Answer is:

So I asked Flightradar about it and they sent me the response below:


  Hello Jean,
  
  Thank you for contacting Flightradar24.com.
  
  Your Support reference is 219752.
  
  Yes, that is correct. We provide this free equipment to anyone in the
  world who has a good location with steady internet and power. The
  reason why you don't see many aircraft in the middle and west of China
  is because of the lack of receivers in these area. 
  
  You make an application and we go from there if it's a good location.
  If we already have receivers in the area then we put the application
  on hold but if its a good location then we provide them with a free
  kit. 
  
  https://www.flightradar24.com/apply-for-receiver
  -- Best Regards, Muazzam  Flightradar24


Straight from the horse's mouth the answer is simple "lack of receivers" what causes that is open to speculation but a quick search for "rural china power grid" and "rural china internet coverage" points out "steady internet and power" can be the problem.