My home country Taiwan's parliament is sometimes infamous for parliament brawls which makes headlines on the media. Were there ever cases of such brawls in the history of US Congress or the British parliament?

One example from US history is the caning of Charles Sumner in 1856 (Source):

On May 22, 1856, the "world's greatest deliberative body" became a combat zone. In one of the most dramatic and deeply ominous moments in the Senate's entire history, a member of the House of Representatives entered the Senate Chamber and savagely beat a senator into unconsciousness.

The inspiration for this clash came three days earlier when Senator Charles Sumner, a Massachusetts antislavery Republican, addressed the Senate on the explosive issue of whether Kansas should be admitted to the Union as a slave state or a free state. In his "Crime Against Kansas" speech, Sumner identified two Democratic senators as the principal culprits in this crime—Stephen Douglas of Illinois and Andrew Butler of South Carolina. He characterized Douglas to his face as a "noise-some, squat, and nameless animal . . . not a proper model for an American senator." Andrew Butler, who was not present, received more elaborate treatment. Mocking the South Carolina senator's stance as a man of chivalry, the Massachusetts senator charged him with taking "a mistress . . . who, though ugly to others, is always lovely to him; though polluted in the sight of the world, is chaste in his sight—I mean," added Sumner, "the harlot, Slavery."

Representative Preston Brooks was Butler's South Carolina kinsman. If he had believed Sumner to be a gentleman, he might have challenged him to a duel. Instead, he chose a light cane of the type used to discipline unruly dogs. Shortly after the Senate had adjourned for the day, Brooks entered the old chamber, where he found Sumner busily attaching his postal frank to copies of his "Crime Against Kansas" speech.

Moving quickly, Brooks slammed his metal-topped cane onto the unsuspecting Sumner's head. As Brooks struck again and again, Sumner rose and lurched blindly about the chamber, futilely attempting to protect himself. After a very long minute, it ended.

Bleeding profusely, Sumner was carried away. Brooks walked calmly out of the chamber without being detained by the stunned onlookers. Overnight, both men became heroes in their respective regions.

Another example would be the violence of Nazi deputies in the Reichstag, which was an extension of their deliberate policy of provoking violence in the streets. Benjamin Carter Hett says that “Nazis made a point of violating every principle of parliamentary courtesy.” The street fights and parliamentary brutality were useful political tactics for an authoritarian party trying to muscle and persuade its way into power: the government’s “inability to keep the peace corroded the democratic state’s legitimacy in the eyes of many of its citizens. The growing pervasiveness of violence also prepared people to accept the state violence the Nazis would later direct.”

After the "last election" on March 5, 1933, the Nazis convened a new Reichstag session at the Kroll Opera House. Hitler demanded that the Reichstag pass an Enabling Act granting him sweeping powers. It took enormous physical courage for the Social Democrats even to show up for this session -- they had to run a gantlet of SA intimidation to enter the chamber, and endure physical threats from SA members all around them during the session, but they stood firm against the Enabling Act nonetheless. So in this case, it's not an actual fight, but it is pretty serious physical intimidation of deputies for political purposes. (Source: Hett, Benjamin Carter. The Death of Democracy: Hitler's Rise to Power and the Fall of the Weimar Republic. New York: Henry Holt and Company, 2018.)