RWY INCURSION AS A CRJ200 PIC TURNS OFF RWY 4 ONTO RWY 13 THINKING IT WAS TXWY P AS CLRED BY LCL CTLR AT LGA.

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WE WERE EXECUTING THE ILS 4 APCH TO LGA WITH THE FO FLYING. AFTER TOUCHDOWN AT APPROX 50 KTS; ACFT CTL WAS XFERRED TO ME. TWR INSTRUCTED US TO TURN R AT TXWY P AND CONTACT GND. NOT REALIZING MY ERROR AS I WAS ATTEMPTING TO SLOW THE ACFT TO MAKE TXWY P; TURNED R. SECONDS LATER; I REALIZED I HAD TURNED ONTO RWY 13. REALIZING THIS ERROR; I IMMEDIATELY TURNED R ON TXWY R; AND THEN JOINED TXWY P. WE CONTACTED GND CTL AND TAXIED TO THE RAMP (BALLAST FENCE) AS INSTRUCTED. NO COMMENTS WERE MADE BY EITHER TWR CTL OR GND. I FEEL THAT I SHOULD HAVE INITIATED A XFER OF CTLS EARLIER ON THE ROLLOUT SO THAT I COULD HAVE DECELERATED THE ACFT MORE QUICKLY. I DON'T RECALL IF WE ANSWERED ATC'S INSTRUCTIONS TO TURN R AT TXWY P; BUT MY OTHER CONCERNS WERE THE ACFT ON FINAL BEHIND ME. IN HINDSIGHT EVEN THOUGH I WAS NOT FLYING THE APCH AND LNDG; I SHOULD HAVE MORE THOROUGHLY BRIEFED THE EXIT POINT AFTER TOUCHDOWN AND TAKEN CTLS SOONER. THAT MIGHT HAVE SHORTENED OUR ROLLOUT AND HENCE NOT HAVE OVERRAN THE INTXN P. SUPPLEMENTAL INFO FROM ACN 623483: TWR DIRECTED US TO EXIT ONTO TXWY P AND CONTACT GND. I READ BACK THE INSTRUCTIONS NOT REALIZING THAT WE HAD ALREADY PASSED TXWY P. WE MADE THE FIRST R TURN AVAILABLE TO US; ASSUMING IT WAS TXWY P. TWR MAY HAVE CONFUSED US BY GIVING US TAXI INSTRUCTIONS THAT WE WERE NOT ABLE TO TAKE. REGARDLESS; A MORE THOROUGH TAXI BRIEFING COULD HAVE BEEN DONE BY THE CAPT AND I TO PREVENT THE LACK OF SITUATIONAL AWARENESS.