Come September, when the U.S. women's gymnastics team is expected to place no better than fourth at the summer Olympics, the gymnastics community won't need George Steinbrenner to head a special investigation to find out why.

It's not that U.S. gymnasts lack the skills to earn at least a bronze medal. Rather, gymnastics experts say it's the fighting among the nation's top coaches and the United States Gymnastics Federation that stands in the way."We're fighting with each other more than being concerned with who we are competing with," says one coach.

Some examples of the in-fighting in gymnastics:

- After the World Gymnastics Championships in October, U.S. Olympic Coach Greg Marsden quit because he says the gymnastics community undercut his efforts to do the job right.

- In the last year, Bela Karolyi, who gave the world Nadia Comaneci before defecting to the United States and coaching Mary Lou Retton, has been accused of trying to take control of the sport.

- It was only in March, in what some consider an 11th-hour decision, that the United States Gymnastics Federation picked a new Olympic coach - Don Peters, the 1984 Olympic coach.

Coaches agree that the private club system in the United States is largely responsible for the problems.

In Eastern bloc countries, the world's top producers of gymnasts, the governments build training centers, hire the coaches and reward the families of gymnasts with fancy apartments and cars. A gold medal for them is a victory for the communistic way of life.

"They are . . . more nationalistically prepared," says Mike Jacki, executive director of the U.S. federation.

In the United States, the federal government lends little support. Gymnasts, now numbering 180,000, train at approximately 2,300 private clubs. Each club must build its own facility, pay a coaching staff and assemble a board of directors. In addition, each club is solely responsible for raising enough money to keep itself financially afloat.

At national meets, gymnasts don't represent the United States, they represent the Parkettes, SCATS or Karolyi's. A first-place finish for a Parkette is a victory for her club.

It's only when they compete internationally as members of the U.S. national team that gymnasts drop their allegiances to their clubs. At times, club coaches have made the transition difficult. "I have found the athletes to be cooperative and willing to represent the United States," says Marsden. "The problem that I have run into is when the adults get involved, when the personal coaches get too involved."

Private coaches like Donna and Bill Strauss of the Parkette National Gymnastic Training Center in Allentown admit they are reluctant to turn over their gymnasts to another coach, even for the Olympic team. And, they don't think the U.S. federation understands the reason behind their reluctance.

Under International Gymnastics Federation rules, private coaches are not permitted on the gym floor during team competition. For a coach to be reduced to little more than a spectator can be a trying experience, especially if he or she has spent years developing the gymnast's skills.

"Oh, it's terrible. I'd almost rather not be there," says Donna Strauss.

Their spectator status, however, hasn't stopped private coaches from at least trying to play an integral part in coaching their girls at international meets.

At the 1984 Olympics, Karolyi got around the rules by securing a floor pass that allowed him to move equipment.

This gave him better access to Retton, who won a gold medal at the Los Angeles games.

Until the meet starts, the private coaches have been known to corner the head coach, advising him how to train the gymnast.