The June 1775 battle that ended in a British victory despite heavy losses was the Battle of Bunker Hill.
Fought on June 17, 1775, during the Siege of Boston, the battle was one of the first major engagements of the American Revolutionary War. Colonial forces occupied the Charlestown Peninsula and built a fortified redoubt on Breed’s Hill, close to Boston. British troops attacked the position in repeated assaults.
The British eventually captured the colonial defenses and forced the colonial army from the peninsula, making the engagement a British victory. However, British casualties were severe: the British side suffered 226 killed and 828 wounded, while the colonial forces reported 138 killed, 305 wounded, and 30 captured.
The name can cause confusion because most of the fighting took place on Breed’s Hill rather than Bunker Hill. Bunker Hill was the original objective and gave the battle its commonly used name. The battle also inspired the later Bunker Hill Monument in Charlestown, Massachusetts.