OMA LCL CTLR EXPERIENCED OPERROR WHEN DEP FROM RWY 14R CONFLICTED WITH LNDG ON RWY 18.

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I WAS WORKING LCL CTL AT OMAHA ARPT. I WAS VERY BUSY WITH A LOT OF THINGS GOING ON SIMULTANEOUSLY. RWY 14L WAS CLOSED. ALL ACFT WERE USING RWY 18 OR RWY 14R. I HAD AN ACFT IN CLOSED TFC; TOUCH-AND-GOES ON RWY 18; A HELI TRANSITIONING MY AIRSPACE AS WELL AS AN ACFT WITH A GEAR PROB. AT XA39 I TOLD ACR Y TO 'HOLD SHORT OF RWY 14R.' AT XA41:40 I ASKED ACR Y IF HE COULD MAKE AN IMMEDIATE DEP; HE SAID YES; SO I SAID 'ACR Y CLRED FOR IMMEDIATE TKOF RWY 14R; TFC 3 MI FINAL LNDG RWY 18.' THE WIND WAS ALMOST DIRECTLY OUT OF THE S 180-210 DEGS AT 10-12 KTS. THE ACFT ON FINAL WAS A PC-12/L TURBO PROP; FAIRLY SLOW. ACFT X AS I RECALL. HE WAS DOING ABOUT 110 KTS ON FINAL. AT XA42:33; ACFT X ASKED IF HE SHOULD HOLD SHORT OF RWY 14R; I SAID AFFIRMATIVE. I THEN REALIZED THAT ACR Y WAS JUST STARTING HIS TKOF ROLL 54 SECONDS AFTER HE SAID HE COULD MAKE AN IMMEDIATE DEP. WE CAN'T USE 'POS AND HOLD ANYMORE;' SO I FEEL THIS ACR Y WAS A DIRECT CONTRIBUTOR TO THIS ERROR. GRANTED I SHOULD HAVE SEEN IT; BUT I WAS ALSO WORKING TO KEEP AIRBORNE ACFT APART. SUPPLEMENTAL INFO FROM ACN 730561: I WAS INBOUND FOR LNDG AT OMA. VISIBILITY WAS ABOUT 7 MI WITH CLR SKIES. RWYS IN USE WERE RWY 14L; 14R; AND 18. APCH ASSIGNED ME RWY 18. I JOINED THE LOC AND WAS HANDED OFF FROM APCH TO TWR CTL AT THE MARKER. UPON LNDG; I PLANNED TO CLR THE RWY WITH A L TURN AT TXWY C. NOTE: RWYS 18; 14R AND TXWY C ALL INTERSECT AT THE SAME POINT. I TOUCHED DOWN ON THE TOUCHDOWN ZONE. ON THE ROLLOUT; SOMETHING DIDN'T FEEL RIGHT AND I SLOWED AS I APCHED THE RWY 14R/C INTXN. AS I SLOWLY APCHED AND STOPPED SHORT OF THE RWY 14R/C INTXN; SOMEONE KEYED THE MIKE AND I HEARD 'IS THAT GUY GONNA HOLD SHORT?' 2 OR 3 SECONDS LATER; AS I ATTEMPTED TO LOOK OVER MY R SHOULDER FROM THE L SEAT; ANOTHER VOICE CAME ON FREQ WITH 1 WORD 'JEESUS.' WITHIN 3 SECONDS OF THAT XMISSION I OBSERVED A CRJ TAKING OFF RWY 14R FROM MY R TO L. THEY OVERFLEW THE INTXN DIRECTLY AHEAD OF ME AT AN APPROX ALT OF 70-100 FT. I REMAINED STOPPED ON RWY 18 SHORT OF THE RWY 14R/C INTXN FOR 5-7 SECONDS AT WHICH POINT THE TWR CTLR XMITTED 'ACFT X; TURN L AT TXWY C; NO DELAY....' I RESPONDED 'L ON CHARLIE.' I PROCEEDED IN TO THE INTXN; TURNED L ON TXWY C AND TAXIED TO AND HELD SHORT OF RWY 14L. AFTER 5-10 SECONDS OF HOLDING; I XMITTED 'IS ACFT X CLRED TO CROSS ONE-FOUR L?' TWR RESPONDED 'ACFT X; CROSS ONE-FOUR L; CONTACT GND POINT NINER.' I CROSSED THE RWY AND TAXIED TO THE FBO. AFTER SHUTDOWN; AND UPON REFLECTION; I WONDERED 'WHY IN THE WORLD DIDN'T I HEAR THE TWR CLR THAT JET FOR TKOF ON RWY 14R AND CATCH HIS MISTAKE?' I MUST HAVE HEARD THE TKOF CLRNC WHILE ON SHORT FINAL AND FAIL TO PROCESS IT. ON THE OTHER HAND; DURING MY ROLLOUT; I HAD A STRONG FEELING THAT 'SOMETHING JUST WASN'T RIGHT' AND WAS SLOWED DOWN AND NEARLY STOPPED BEFORE THAT FIRST XMISSION OF 'IS THAT GUY GONNA HOLD SHORT.' IMPORTANT NOTES: I WAS CLRED TO LAND ON RWY 18. I WAS NOT ISSUED AND DID NOT ACCEPT A LAHSO CLRNC. FACTORS THAT CONTRIBUTED TO MY FAILURE TO RECOGNIZE THE CTLR'S MISTAKE AND THE CONFLICT IT CREATED: CUMULATIVE FATIGUE -- I WAS 11+ HRS INTO MY THIRD LONG DAY IN A ROW. I WAS ON MY 12TH OF 13 LEGS IN THE PREVIOUS 3 DAYS INCLUDING MULTIPLE INST APCHS. I WAS SINGLE PLT FOR ALL OF THESE FLTS. THE CURRENT ATIS AID 'LNDG AND DEPARTING RWYS 14L; 14R AND 18.' THERE WAS NO SPECIAL NOTE SUCH AS 'OPS TO INTERSECTING RWYS ARE IN USE.' ANY TIME OPS INVOLVING INTERSECTING RWYS ARE IN USE -- ESPECIALLY IN VMC -- THE POTENTIAL FOR CATASTROPHIC TFC INCURSIONS EXISTS.