For example I have a website where advertisers ads are running... Its not a matter for me if visitors are clicking on those ads or not but is there any way that advertisers ONLY pay for the visitors they get from our site?

If they get 1000 visitors from my site they will pay $10
If they get 2000 visitors.... will pay $20...etc

is there any site that does so?

What you think is it good idea of advertising? I think yes, cause advertisers will get 100% results of what they are investing.

This seems to be a very good opportunity of earning. Which company provides this? You didn't mention any name or URL there.
I am interested to become an affiliate of that site.