"You only have to look at what happened to Vijay Singh just recently to know the drugs issue is there," Norman told The Australian newspaper in comments published on Monday.
The Fijian told an American magazine in January that he had been taking deer-antler spray as a supplement, saying he was unaware it is believed to contain the hormone IGF-1, which is a banned substance. It can only be detected by blood tests.
"How deep it is (the problem), I have no idea because we only do urine analysis instead of blood testing," Norman added. "If you really want to be serious about it and find out about what's really going on, we need to do blood testing. I think it's disgraceful, to tell you the truth. The golf associations have to get together and step it up.
"It's a pin prick for a player and you find out what's going on. If you're the head of golf or any sport, if you're the commissioner for a sport, it's your responsibility to make sure your sport is clean. That should be your No 1 priority.
"Any sportsman or sportswoman who uses an outside agency to improve their skills is cheating. It sickens me. They're putting a black eye on their sport. If a sport gets itself clean, the corporate dollars will always be there because people will know it's a sport they can trust. The rest will take care of itself."
R&A chief executive Peter Dawson, speaking at a press conference ahead of this year's Open Championship at Muirfield today, said: "I can't speak from the Tours' position on this and week-in, week-out of course it's the Tours who are administering drug testing.
"As far as Olympic golf is concerned, when the players come under the International Golf Federation policy that will be a different story.
"We have a medical team working at the IGF now looking at golf's testing regime and coming up with recommendations for what's going to happen pre-Olympic games. So the issue of blood testing is up for consideration at the moment."