Bitter street fighting between riot police and supporters of the suspended Solidarity trade union engulfed Warsaw and other big cities today in what could prove a turning point in Poland's political and economic crisis.

The clashes in Warsaw erupted after police, using water cannons and tear gas, broke up a peaceful demonstration by tens of thousands of young people in the Old Town to protest martial law. The rioting spread to other parts of the city and at one point during the late afternoon raged around the Communist Party headquarters near parliament where deputies were discussing government proposals for national reconciliation.

Telephones were cut off throughout the city two hours after the demonstrations began at 4 p.m. and the martial law authorities warned that a night curfew, lifted only yesterday, might be restored.

No official figures were given for injuries or arrests, but ambulances were seen driving in and out of parts of the city where fighting was heaviest. Witnesses reported seeing numerous demonstrators beaten by police wielding rubber truncheons and others taken away in vans under arrest.

Political analysts said that the rioting, the worst outbreak of street violence here since the imposition of martial law last December, could have major political significance. It demonstrated that public hostility to the military gpvernment still runs very deep and is liable to erupt into the open as soon as rigid controls are relaxed.

Some senior Solidarity advisers said they fear the street fighting could strengthen the position of hard-liners within the Communist Party apparatus who favor a total ban on the union. It could undermine the position of the martial law leader, Gen. Wojciech Jaruzelski, and his relatively moderate faction within the party.

Today's clashes in Warsaw followed a peaceful demonstration on May Day by well over 20,000 Solidarity supporters who called for the lifting of martial law and the freeing of an estimated 2,000 detainees. Underground Solidarity activists had called for further demonstrations to be held today before a mass marking the anniversary of the granting of Poland's first democratic constitution in 1791. The government had issued warnings that all unofficial gatherings were illegal under martial-law regulations.

Undeterred by a heavy police presence set up at the principal entrance to the old part of the city, thousands of people began gathering outside St. John's Cathedral in the Old Town. A huge cheer went up as the first Solidarity banners were unfurled above the heads of the crowd in the narrow street. There was a swell of applause when a poster of the union's detained leader, Lech Walesa, was plastered onto the outside wall of the cathedral.

"Lift martial law" and "Free Lech Walesa," the crowd chanted as hundreds at the end of the packed corridor began surging toward the royal castle, the symbol of Poland's independent nationhood. It was there that the riot police had set up their lines, and the first serious clashes took place as police attempted to disperse the demonstrators with water cannons, tear gas and repeated baton charges behind reinforced plastic shields.

As police units began to surround the Old Town, parts of the growing crowd were pushed into its maze of narrow streets and out into the city. Other demonstrations seemed to have risen up spontaneously in other parts of Warsaw.

Fighting raged for six hours around the city. In the principal confrontation in and around Old Town, crowds chanting "Gestapo, Gestapo," responded to the police onslaughts by hurling rocks and tear gas cannisters back at them. Barricades were built during a two-hour battle for control of the Old Town square, which ended only when heavy reinforcements of police were brought in.

The clashes took place in a city still bedecked with red banners and slogans lauding Communist rule in honor of May Day. Youths later ripped red flags down from buildings, trampling them underfoot and burning them.

An official statement read on television news during the evening described the Solidarity demonstration as "a desperate attempt at revenge by the enemies of socialism." The television announcer said that similar incidents had occurred in some other towns, including Gdansk, where Solidarity was born following workers' strikes in August 1980. No details were provided, however.

At a news conference that had been previously scheduled, Deputy Prime Minister Mieczyslaw Rakowski said late this afternoon that he was saddened to hear about the protests.

"There are still some people who want to provoke a confrontation," he said. "But if they think they can achieve their aims, they are making a tragic mistake."

A senior Solidarity adviser, however, blamed the disturbances on the government's inability to develop any coherent political strategy since last December's military crackdown. The adviser, who wished to remain anonymous, said Jaruzelski had failed to come up with any clear vision for ruling Poland without the protective shield of martial law.

Despite informal contacts between government and Solidarity officials, there has been no serious attempt to open substantive negotiations. The mood of political drift has particularly dismayed Catholic Church leaders, who have warned repeatedly of major clashes in the absence of some form of national reconciliation.

While most of the participants in today's demonstrations were young people, they obviously had the support of most Warsaw residents. Some motorists, surprised at tear gas cannisters exploding around them during the evening rush hour, sounded their horns in support. Bystanders chanted abuse at the police, who responded by waving their truncheons.

The police set up a makeshift command post near the royal castle where they brought detained demonstrators. People arrested were thoroughly searched and Solidarity badges ripped off their jackets.

A witness said he saw two plainclothes policemen kicking and beating a student who yelled back: "You're Gestapo, you're worse than the Nazis." After kicking him in the ribs and chest and removing his belongings, they ordered him to go home.

A woman muttered under her breath: "Do these people really think they are the protectors of the working class?"

Although much of the fighting broke out spontaneously, the initial demonstration was organized by the same people who led the Solidarity march through Warsaw on May Day. One of these leaders, a bearded student with "Solidarity" emblazoned across his T-shirt, was arrested after being plucked from the crowd by a plainclothes agent.

By midnight, the streets were quiet and patrolled by long convoys of police trucks, jeeps, and water cannons. The pall of tear gas hung over the city.

A planned broadcast by Radio Solidarity announced for this evening failed to take place. Shortly before 9 p.m., government stations started broadcasting loud music on several empty wave lengths, thus drowning out the weak Solidarity transmitter.