RWY INCURSION AT EWR; NJ.

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AFTER CLRING RWY 4R AT EWR; TWR INSTRUCTED US TO TAKE TXWY P AND STAY WITH HIM. I HAD EXITED RWY 4R AT HIGH SPD TXWY L AND HAD MADE THE L TURN ON TXWY Y WHEN INSTRUCTIONS WERE DELIVERED. TXWY P IS PARALLEL TO RWY 4L/R AND BTWN THE 2 RWYS. NOT BEING FAMILIAR WITH THE LOCATION OF TXWY P AND EXITING THE RWY IN THE IMMEDIATE PROX OF IT; I OVERLOOKED THE TXWY MARKER. I WAS ADDITIONALLY DISTR BY BRIGHT CONSTRUCTION LIGHTS ADJACENT TO RWY 4L AND 500-600 FT TO MY L. ASSUMING RWY 4L WAS CLOSED AND TXWY P WAS AHEAD OF ME; I CROSSED RWY 4L. NO ACFT WERE ON THE RWY AS WE CROSSED. TWR ADVISED US OF OUR ERROR AS WE CLRED RWY 4L. THIS ERROR OCCURRED DUE TO MY EXPECTATION THAT I WOULD BE CLRED TO CROSS RWY 4L AT TXWY Y NOT IMMEDIATELY KNOWING THE LOCATION OF TXWY P AS I EXITED RWY 4R; AND MY MISTAKEN PERCEPTION THAT RWY 4L WAS CLOSED AND UNDER CONSTRUCTION (AS INDICATED BY BRIGHT CONSTRUCTION LIGHTS DOWNFIELD). RWY 4L WAS NOTAMED 'LAST +/-800 FT CLOSED FOR LNDG;' BUT WAS AN OPERATIONAL RWY. INCREASED ATTN TO ALL POSSIBLE TAXI RTES FROM LNDG RWY WILL BE MANDATORY IN MY ARR BRIEFINGS. SUPPLEMENTAL INFO FROM ACN 477427: I READ BACK THE INSTRUCTIONS AND THE CAPT CALLED FOR THE AFTER LNDG CHK. I WAS NOT FAMILIAR WITH THE ARPT AND CHKED MY ARPT DIAGRAM AND NOTED TXWY P WAS A QUICK TURN TO THE L AND BTWN THE RWY 4R/L. I KNEW THE CAPT WAS FAMILIAR WITH THE ARPT AND THOUGHT HE WAS COMFORTABLE WITH THE TAXI INSTRUCTIONS AND THEREFORE CALLED FOR THE CHKLIST. I WENT 'HEADS DOWN' WHILE ACCOMPLISHING THE CHKLIST AND WHEN I LOOKED UP WE WERE XING RWY 4L AT TXWY Y. EXITING THE RWY IS A BUSY TIME IN THE COCKPIT; PARTICULARLY ON A HIGH SPD. YOU ARE DECELERATING; TALKING ON THE RADIOS; ORIENTING YOURSELF AND ACCOMPLISHING CHKLISTS. I THINK WE MIGHT HAVE BEEN VERY CLOSE TO TXWY P WHEN GIVEN THE INSTRUCTIONS SO THE CAPT WAS LOOKING BEYOND THE TXWY P TO TXWY R; WHICH HE WAS ACCUSTOMED TO TAXIING ONTO THE PARKING AREA. THE LAST 810 FT OF RWY 4L WAS CLOSED AND THERE WAS NO CONFLICT WITH TFC. I BELIEVE IF THE CTLR WOULD HAVE MENTIONED 'HOLD SHORT OF RWY 4L' IN HIS CLRNC; WE WOULD NOT HAVE CROSSED IT; EVEN AFTER MISSING THE TURN. WE MISSED THE TURN AND DIDN'T CATCH IT UNTIL IT WAS TOO LATE. IF THE TAXI INSTRUCTIONS WERE MORE COMPLETE (IE; 'HOLD SHORT OF RWY 4L AT ___') THE INCURSION WOULD NOT HAVE OCCURRED WITH US MISSING THE TURN. SUPPLEMENTAL INFO FROM ACN 477757: I DO NOT HAVE A TXWY DIAGRAM SO ASSUMED; AFTER COMPLETING AFTER LNDG CHKLIST AND COMMUNICATING WITH THE COMPANY AS TO OUR GATE; AS WE TAXIED ACROSS RWY 4L THAT TXWY P WAS ON THE OTHER SIDE.