Is there 4 or 5 major sports in America?
Yes. At least that many.
Define both major and sports. For many football is at least two different sports, college and NFL. Basketball might be three or four, College and pro for both men and women. Hockey would not be major around here but probably only pro. Baseball is similar, college, professional and MBL. So it could be many.
NBA, MLB, NFL, NHL
To some people there are four Football, baseball, basketball & hockey, many others see soccer as a major sport and to them it's five.
Football, baseball, basketball, hockey... Four. By the way, it's "are," not "is."