Complex sentence: With the beat boom in 1963, Liverpool became the center for British rock, but the grittier London bands like the Rolling Stones quickly established themselves as a part of the British scene and were second in popularity only to The Beatles.
Split into simpler sentences: With the beat boom in 1963, Liverpool became the center for British rock. However, the gritty London band the Rolling Stones, quickly established themselves as a part of the British scene and were second in popularity only to The Beatles.