I've tutored math for many years, both as an employee of a tutoring center and independently. What materials I use depends on the purpose of the tutoring.

Most often, you are hired because a student is not getting the grades in math that the parents want. So you work with the student on the materials the classroom teacher provided. I'm not sure what you mean by

There is no concept of a "textbook" in the US."

On the contrary, it is almost unheard-of in the US for math to be taught without a textbook. Furthermore, the teacher never assigns all the problems in the textbook, so you have plenty of extra practice material.

It often turns out that the student is doing poorly because of some gap in prerequisites--doesn't know the times tables, doesn't understand fractions, etc. Then I use workbooks that I own, that I bought online. Some I like are the Key To... series from Key Curriculum Press; there are sets for fractions, decimals, percentages, geometry, etc. Their layout is informal, which helps the student not feel threatened. (Sometime the job is not about teaching math, but about giving the student confidence to overcome math anxiety.) I also have an "algebra bootcamp" workbook.

If I'm hired to tutor specifically to prepare for an exam like the SAT or AP, I have the student buy a test-prep book from Princeton or Barrons and we use that. For the APs, there are also lots of free-response questions on the College Board site with grading rubrics.

On the rare occasions when I have been hired to enrich the mathematical universe of a bright and interested student, I've used competition math books from the Mandelbrot contest and Art of Problem Solving.