The D.C. government said yesterday it has delayed mailing 17,900 water bills because it incorrectly read water meters, failed to repair broken meters and improperly programmed its billing computer.

Moreover, city officials said they are receiving an average of 3,750 complaints a month from city residents who say they have been sent grossly exaggerated bills, had their water shut off even though they paid their bills on time or encountered other billing problems.

The 17,900 "problem accounts" for which bills have not been sent represent more than 15 percent of the city's 112,000 commerical and residential water customers. Nonetheless, William B. Johnson, director of the city's Department of Environmental Services, told a City Council committee yesterday that the city has recently made "substantial progress" in its water billing operations and that it hopes to eliminate the problems by the end of the year. He pleaded with the council for "patience and understanding."

But several council members said their patience has been exhausted. The city's claims of progress also failed to satisfy Margot Higgins of Capitol Hill, one of several witnesses who described for the committee what she called "horror stories" about water bills.

Higgins said she is a widow who lives alone in a small house. She said last August her water bill inexplicably jumped from $26.61 to $244.46 for a five-month period in which she was away for one of the months. She was given a deadline for paying the bill that expired yesterday, but she did not pay it.

"I didn't become a whale, I haven't flooded Northeast to become a skating rink and I don't take 13 showers a day," she said. "People in this city are starting to feel that the water board is out to get them using a Russian roulette system."

Council members said they have recently been deluged with hundreds of similar complaints from angry constituents.

"This problem has been causing a greater amount of unrest in our community than anything else," said Councilman Jerry A. Moore Jr. (R-At-large), chairman of the Transportation and Environmental Affairs Committee.

Moore told Johnson, "In my opinion, I've given you all the time you need to straighten it out. I'm at the point now where I can't listen anymore with an ear of patience. Something has to be done about this and it has to be done now."

Council member Betty Ann Kane (D-At-large) told Johnson his statement was "filled with the same words we've been hearing over and over again now, year after year -- we plan, we hope, we intend. I don't see anything that will restore customer confidence that the numbers on their water bills bear any relation to reality.

"I've had it and my constituents have had it," said Kane, who added that the city recently mistakenly sent her a water bill that was 50 percent too high.

Kane said she would soon introduce a bill that would end the city's "water mess" by eliminating individual meter readings and charging all residential customers a flat rate based on average citywide residential water usage. She estimated that the flat rate probably would be "about $150 a year," slightly less than the $168 current average for a family of four.

The political overtones of the water issue were underscored yesterday when Mayor Marion Barry made an unusual appearance at the committee hearing to listen to the testimony and occasionally coach his aides on how to respond to council members' questions.

He suggested to a reporter that his political critics were trying "to make hay out of this issue" and derided Kane's bill as inequitable. "Why should I pay for my neighbor's bill on an average basis if he uses more water than I do? That's crazy."

When he was running for mayor in 1978, Barry criticzed former Mayor Walter E. Washington's inability to solve the water billing problem. Barry, who is expected to seek reelection next year, emphasized that he intends to make sure that the water problems will be "99 percent" solved by next spring.

"Johnson's behind -- and City Administrator Elijah B. Rogers' behind -- are on the line on this one," Barry said. "I've told them that."

The inability to send out accurate and timely water bills has been a continuing problem that the city attempted to resolve more than 2 1/2 years ago when it transferred a manual bookkeeping system for water accounts onto computers. But Johnson said yesterday those manual records were riddled with errors and omissions, causing "much bad data to be shoved into the computer."

In a further effort to correct the problems, the city signed a $243,000 contract with a Dallas-based firm, Optimum Services Inc., to operate the water billing system. Last summer, the city asked a management team from Potomac Electric Power Co. to study the water system.

Barry said the city has implemented all but one of 21 management recommendations by the Pepco managers. Other city officials also contended that one reason for a recent upsurge in complaints is that it has begun to cut off water service to customers who have been delinquent in their bills or refuse to pay them.