Question: Recently I’ve got stumbled upon the question: What’s the difference between *quite a lot*, *quite a few*, and *quite a bit*?

This is very confusing considering *a lot* and *a few* have almost opposite meanings.

As far as I can see in dictionaries *quite a <COUNT-WORD>* is always *many*. Is that so, or am I missing some important difference?

Answer: 
A few only with countable nouns. Generally a low quantity.
A few people helped me finish.

A bit of only with uncountable nouns. Low quantity.
A bit of help goes a long way

A lot of with either countable or uncountable nouns. High quantity.
A lot of people know me!

Now by adding quite to these, they reduce the idea of a lot, and they increase the ideas of a few and a bit.

Quite a few people helped me finish. Now it's a good quantity.
Quite a bit of help goes a long way. Now also a good quantity.
Quite a lot of people know me. Now this is not as emphasized as before, it has become a little softer.

Hope this helps!
Also, I would probably say "There are *quite a lot* of words on the page", but "There are *quite a few* words on the page" also sounds correct. However, "quite a bit" doesn't work in the same context.