Is it true that students from European countries have had many more advanced courses in mathematics when they arrive to Grad school in the US? This question is inspired by what Steven Krantz wrote in his book "A mathematicians survival guide." 


  If you are an American student, educated in the USA, then you may 
  find that the international students in your graduate program appear to be 
  way ahead of you. First of all, students in Italy, France, and other European 
  countries often specialize much earlier than we do in the United States. They 
  have had many more advanced math courses than you will have had. [I was 
  once told that a typical Italian math student graduating from college will 
  have taken 22 year-long math courses. I have not had 22 year-long math 
  courses in my entire life.] 


If this is not true, then in your experience, from which countries/universities do you get the most advanced graduate student?
First and foremost, it is a important to note that there is a vast amount of variation between institutions and individuals, so generalizations are not terribly helpful here.

That said, historically there have indeed been differences in the philosophy and organization of European and American approaches to education. European education has tended to put more emphasis on knowledge, where American education has tended to put more emphasis on creativity and communication.  As a result, many European students are already specializing in high school, while many American students are required to remain generalists until university and may in fact still be taking classes outside their "focus" all the way through the end of their undergraduate studies.

This does not, however, provide any clear advantage to one group or the other when it comes to doing a Ph.D.  At the doctoral level, one needs both knowledge and also creativity and the ability to communicate well with colleagues. Thus, while students educated in one tradition are more likely to enter graduate school being able to prove Urysohn's Lemma, students educated in the other tradition are more likely to be able to effectively communicate why that proof is interesting.

In short: the word "appear" is in the original quote for a reason.