Finding where a page / site ends in Google for a specific search query

I know this is a question that has been asked a hundred times on the internet, but I have several reasons to ask it here: (1) so this is on SO; (2) it's hard to find - a lot of rubbish.

What I'm trying to figure out is which page like example.com is in google search results like "stackoverflow.com". Is there a website that does this and is reliable?

I understand that it differs depending on what browser, country, language, etc. and so on, but perhaps in the end the site even provides a link to the page where the result is included so you can determine what it was looking for.

At the moment, the best method I've found is to press next, ctrl + f, enter, next ... And abandon page 5 or so, because someone looks so far away.

Edit: I am not looking for this search query stackoverflow.com site: example.com ". My search string is" stackoverflow.com "and I am trying to find example.com in the results.

The reason I am asking & the reason this programming is related: The reason I am asking is because it has to do with SEO. I often have clients who want to know where they are in the search results after some HTML changes they paid for.

Webmaster Tools: Yes, Webmaster Tools are a great tool for finding keywords that do well, but I find that when you first start out you can be very, very far off the list and they just don't show it ... I'm talking on page 15. So you start the website, it's on page 50, you want to know that next week it was up to 40, and the changes you made made a difference.