When Tom Bradley bought a house in the white neighborhood of Leimert Park in 1950, he hung back and let a white friend negotiate the purchase. That "was the way it was done," Bradley said. "Getting mad wouldn't have got us our house any faster."

Nearly three decades later, as mayor of the city, his black and liberal political allies sought his support for a school busing plan that had enraged white residents of the city. Again he hung back. "I could not ask people to obey the law," he said, "if I did not appear impartial."

In this city of confrontation artists and drama buffs, stories abound of the caution of Thomas Bradley, a hulking, subdued, 63-year-old expoliceman on the verge of becoming a national political figure -- and even more of an anachornism. The man who cringes at being called a black politician is now one of the most important black politicians in America.

Within a month, Bradley is expected to be elected to a third term as mayor of the nation's third-largest city, facing only a last-grasp challenge from former mayor Sam Yorty. All other viable contestants have cowered at Bradley's polictical magic.

A poll shows him the most popular politician in California, and by next year, his friends are saying, he has a chance of becoming the first black ever elected governor of one of the states.

He is the very opposite of the ideological gadfly he is being touted to succeed, Edmund G. (Jerry) Brown Jr., as governor. One admirer called Bradley "not a creative thinker nor an activist."

In an interview, Bradley said the Democratic Party does not need a new message to reserve the electoral tide toward the Republicans as much as it needs a better focus on its proven ideas and old coalitions.

"If people perceive the social programs of the Democrats are not careless spending but essential, if they don't perceive things in a negative fashion," Bradley said, then the party could win elections.

How that vague approach will help is unclear, but Bradley has made it work marvelously well. He is a campaign manager's dream, respected by conservatives for his pro-business views and personal reserve and trusted by liberals because of his instinctive grasp of the needs of the poor and because of (although he would deny it) his race.

Talking in a office decorated with Japanese artifacts, Bradley said the growing color-blindness of the electorate is "one of the most heart-warming developments that has taken place in the city."

But his race and upbringing remain crucial in a city in which blacks, Hispanics and Asians now make up a majority of the population. "I like Tom Bradley," a liberal voter mused, "but I wonder if I would like him as much if he were not black."

Bradley's solid conservative approach, his stubbornness, have brought him everything. Beaten in his first mayor's race in 1969, he worked hard as a city council member and came back to win in 1973.

Told that the city's bid for the 1984 Olympics would enrage cost-conscious voters, Bradley pushed forward, and money and facilities for the sports spectacular were found.

Republican businessmen like Philip M. Hawley, of Carter Hawley Hale Stores Inc., openly praise him. "During the drought four years back, he was very even-handed and very thoughtful, asking everyone to change their life style and preserving."

Bradley said people appreciate "the way I've managed this city, a tough executive who doesn't go for nonsense, who is fiscally conservative."

A reporter who watched him visit black churches in the south-central city said he has none of the flamboyance of an Edward M. Kennedy or a Jesse Jackson. He is calm and fatherly, 6 feet 4 and 220 pounds, and "down there he is revered."

Bradley was born in Calvert, Tex., and was brought to Los Angeles when he was 7. His parents separated. He attended UCLA on an athletic scholarship, starring in the quarter-mile, and in 1940 began a 20-year police career.

He earned a law degree at night school, won election to the city council and then as mayor. His wife, Ethel, is a rabid Dodgers fan, attending nearly every home game, and they have two daughters in their 30s.

A recent statewide Field poll gives Bradley a 63 percent favorable rating, compared with 49 percent for Brown, who plans to run for the Senate, and 54 percent for the Republican lieutenant governor, Mike Curb, so far the leading GOP candidate for the governorship.

A Los Angeles Times spoll shows that whether voters favor higher taxes or not, favor gun control or not, they like Bradley although he may be vulnerable to a strong anitcrime candidate.

Bradley reacts to Republican efforts to shift public problems onto private business as if it had been his idea all along.

"I don't think there is a city with better relations with the private sector," he said. But he expects what he calls the "public euphoria" over President Reagan's efforts to cut taxes and spending to change "when they realize how it affects them."

Bradley said he worries about the effects of an imminent end to jobs for 5,200 city workers under the Comprehensive Employment and Training Act.

"A woman who is a clerical employee will be eligible for a two-week vacation on April 4, but she won't get it because she'll be terminated March 31," Bradley said. "Her 4-year-old son is blind. She has no idea where she is going to get a job. She can't pass the civil service test. Her son gets welfare benefits, but it is not enough. It is really a terrible burden for her and many other people."

Businesses were fleeing to the suburbs when Bradley took office, so he initiated a downtown redevelopment plan that persuaded Atlanitc Richfield, Crocker and Wells Fargo banks and other companies to build there.

With federal and private funds, his community development agency built senior-citzen and low-cost housing and helped revitalize some ethnic markets and shopping areas, easing a cosmopolitian capital of western North America.

But Bradley says private business cannot take up the slack of unempolyment caused by federal budget cuts. "I don't think there will be an explosion" of the 1965 Watts variety, he said, "but there will be frustration and anger."

Bradley lost his 1969 race for mayor to the incumbent, Yorty, who was able to manipulate the racial fears generated by memories of Watts.

By 1973, Bradley's reputation for caution had grown, memories of Watts had faded and the percentage of black and Hispanic voters had increased. He won 54 percent of the vote against Yorty. In 1977 he won nearly 60 percent of the vote against state Sen. Alan Robbins and tax-cut activist Howard Jarvis.

Yorty this year, at 71, is a pale ghost of his former rambunctious self. He has called for hiring more policemen, and has accused Bradley of weakening law enforcement. Police, fire and water and power workers' unions have endorsed Yorty, but Bradley's advisers no longer fear the former mayor. They worry only that Bradley might not equal or top his 1977 winning margin and slow his momentum into the 1982 governor's race.

Bradley will say only that a race for governor is "possible." Some Democrats wonder if he can beat a Republican like Curb or Attorney General George Deukmejian, who might label him soft on crime for his efforts to curb some police practices he considered abusive.

Some Democrats are proposing as an alternative candidate former U.S. senator John Tunney, a lawyer and news commentator here, but Bradley remains far ahead in the polls.

If elected governor, Bradley would become an immediate national media star. And although few think he has much interest in national office, he has a history of resisting pigeonholing.

A visitor once called him "a black Gerald Ford, not a black John Kennedy," and he replied, "I'm not a black this or a black that, I'm just Tom Bradley."

Said businessman Hawley: "He is perceived by the average voter as fair and conscientious, as someone who applies himself to the job, and that is a hard combination to beat."