Is there any good tutorial about how to make asymptotic notations rigorously? I think the way from Turing machines to the analysis of pseudocode is quite a big. Or is the situation that there are some material where people learn to use analysis but there is no book where all the details are written out explicitly?
Asymptotic behavior is studied in many math fields such as analysis and statistics. In CS it shows up in complexity theory and the analysis of algorithms. Any good undergraduate book on any of those topics will give you the basis. But don't start with Turing machines. As you note, the path will be very very long and you might never get to what you want to learn.
Even a few hours on wikipedia will teach you a lot. Start with Asymptotic analysis and follow links from there. If you find it hard to read, find a textbook on Calculus to start.