A B757 CREW; BASE TO FINAL FOR RWY 24R AT LAX; WAS UNABLE TO COMMUNICATE WITH SOCAL DUE TO THE CTLR'S TASK SATURATION. THEIR FIRST CONCERN WAS THEY COULD NOT MAKE A TIGHT ENOUGH TURN FOR FINAL WITHOUT FLYING THROUGH THE CTRLINE INTO THE APCH OF A B747. ALSO; THEY SWITCHED TO TWR WITHOUT BEING HANDED OFF SINCE THEY COULD NOT SPEAK TO THE CTLR.

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SOCAL APCH WAS SO BUSY THAT ON 3 OCCASIONS; WHEN PEOPLE CHKED ON FREQ; HE TOLD EVERYONE TO STAND BY. WE WERE CLRED FOR VISUAL AFTER A TURN WITH 90 DEG INTERCEPT TO RWY 24R. NO PRECEDING TFC WAS ADVISED. CTLR WAS LINING ACFT UP TO BOTH RWY 24L&R RWYS FOR SIMULTANEOUS APCHS. TFC WAS ALSO LNDG RWY 25L. WE TURNED FINAL; AT A STANDARD RATE; FROM OUR ASSIGNED HDG AS SOON AS WE WERE CLRED FOR THE VISUAL APCH. THE VECTOR WAS TOO TIGHT AND WE FLEW THROUGH THE FINAL FOR BOTH RWY 24L&R. NO TFC CONFLICT OCCURRED EITHER FOR THE RWY 24 COMPLEX OR THE RWY 25 COMPLEX. HOWEVER; A B747 WAS LINING UP FOR THE PARALLEL RWY 24L. WE DID FLY THROUGH HIS FINAL. CTLR WAS SO BUSY; MANY ACFT (INCLUDING US) SWITCHED OVER TO THE TWR WITHOUT BEING INSTRUCTED TO DO SO. WE WERE INSIDE THE FAF WHEN WE DID. THERE WAS NO GAP ON APCH FREQ TO ASK TO SWITCH. CTLR WAS DOING BEST HE COULD BUT WAS TASK-SATURATED. I DON'T KNOW HOW HIS AIRSPACE BECAME SO FULL. WE HAD A DELAY AND WHEELS UP TIME IN SFO. SUPPLEMENTAL INFO FROM ACN 454410: DUSK LIGHTING CONDITIONS MADE US EXERT WAY TOO MUCH EFFORT TRYING TO SPOT THE NUMEROUS TFC AND MAKING SURE WE WERE FOLLOWING THE CORRECT AIRPLANE.