No power up but turns over good but no start - 1999 Chevrolet Corvette

If no power work your way down start with the battery and make sure is good...you can try jump-starting it if you are not sure...battery cables properly tighten if no power...start checking fuses

I do not understand what you mean. If the engine starts it is turning over. Turning over means the engine turns not that it starts. So the engine cranks but does not start? You need to check the crank position sensor. It should have 3 wires at the connector. 1 is power 12V, 1 is ground and the other is the square wave signal. with a volt meter make sure you have power at the wire closest to the curves side of the connector. Check resistence to ground on the wire closest to the square end of the connector. You cannot check the signal with out a lab scope or graphing multi meter. If power and ground are good you need a mechanic.

Battery good, cables good, all connections clean and tight, no corrosion on battery terminals?
Try to determine if click is from solenoid on the starter or starter relay. Turn the headlights on and watch them while someone turns the key to start. If they get dim, probably the starter-have it tested-even new ones are sometimes bad. If lights don't dim, you will need to check the start circuit. Check for power on the small wire to starter solenoid when key is held in start. If no power there, work back towards the ignition switch. Check the neutral safety switch-should have power in and power out when key held in crank. The ignition switch will have a start wire out of the switch going to the neutral safety switch. It should have power with key in start, the only time the start wire will have power.
Post back with what you find-good luck.

Bad connection at the battery. As soon as large current is demanded by the starter, you lose the connection. Make sure both terminal connections on battery are clean and tight. Now is a good time to do a voltage drop test. Watch youtube videos. Most are quite good at explaining the test. This test is great for finding a bad connection.

Two parts to this circuit, the low current: power supply to the ignition switch which then passes to the small wire on the solenoid when you turn the key to the start position, at this point there should be an audible "click" from the solenoid. Don't forget the solenoid needs an earth to operate, so check the earth path too, usually through its attachment point(s) Check that the ignition switch has power going to it. Check that the power is transferred to the start wire when the key is turned. Check the power is arriving at the solenoid, small wire. Check the solenoid has a good earth. Does the solenoid "click" when you try to start? Yes to all this? Then the low current side is good. The high current side: a permanent live (hot) to one side of the solenoid. When the key is turned to the start position and so long as the solenoid "clicks" then the power to the permanently live terminal is transferred to the other high current terminal. If it is not, then the terminals inside the solenoid are burned/dirty/high resistance. You might be able to dismantle, else change the solenoid. If the power does transfer, but the starter motor does not turn, then suspect the starter motor itself. Check there is a good earth for the starter motor, the heavy duty cable to the engine is in good condition, terminals clean and tight etc. If all good, then the starter motor is most likely U/S.

If by lights, you mean the dash lights come on, then ignition switch must be getting power. If dash is not working with key on, check the 30 amp fuse link which powers the ignition switch.
If switch is good and starter is good, the start wire from switch goes to, if automatic, to inhibitor relay (the park/neutral function) and then to starter. There is also an inhibitor switch on the shifter or on the transaxle that turns on the inhibitor relay when key is on and shifter in park or neutral. The inhibitor switch is powered by a 10 amp fuse D.
If you have a manual transmission, the start wire from ignition switch goes to the clutch interlock relay and then to the starter. This relay is turned on by the clutch interlock switch on the clutch pedal.
You can check for power on either relay, the inhibitor relay or the clutch interlock relay, with a test light. Pull the relay out, have the key held in start position. Now both relays would have power showing on two terminals where the relay plugs in. If inhibitor relay only has power on one terminal or no power at all, either ignition switch is not sending power or fuse D is out or the inhibitor switch is not working right. If clutch interlock switch does not have power on two terminals either the ignition switch is not sending power or the clutch interlock switch is bad.
If the relay checks are good, check for power on the small wire to starter with key held in start. If you have power there, the starter solenoid must be out.

Just concentrate for the moment on the battery, starter, and ignition switch. Your 92 jeep is a complicated marvel, but all you need to do is get the starter turning the engine for all those automatic things to happen and turn it back into a mobile jeep.
As long as you have battery power at the switch, and even bigger battery power at the starter, and you connect the three of them together...instant bingo, the engine starts, or should if everything is working right. If I understand you right, you do have power to the switch, just nothing in start, or you 'd have noted that. Check engine light on, and warning lights and gauges work, yes? Then the switch is working that much, anyway. The switch start position should transmit power to the starter on the small wire that is on the starter. You can check for power there when the key is held in crank (start). Easy to check with a meter or a testlight. You stated battery and starter are good, so if the cable connecting them is good and tight., your problem may be in the start circuit from the switch to the starter (starter solenoid, to be exact). So it's ign. switch to starter relay to starter (solenoid).. Check for power on that path. Quick check at starter relay: pull the relay out (in the power distribution box) it connects to 4 terminals.. One terminal should be hot all the time. check with testlight. If nothing there, check the 40 amp fuse or battery connection or wiring. If one is hot, then have someone hold the key in crank. Now 2 terminals should have power. If not then ignition switch or the neutral safety switch or the wiring between them needs diagnosing. If two terminals do have power, try another relay. maybe yours went out. If a good relay doesn't help, then starter solenoid or starter or your wiring needs further troubleshooting. Hope you find it, over...

make sure all grounds are clean and area of metal that ground wire's bolt to are clean and rust free if that check's out ok there could be a break in a wire going to the starter and most starter's have two wire's going to them and those both need to have power going to them one drop's off when key is turned to starting mode volt meter work's good for this or test light or the starter could be bad

Do you have power?
battery might be drained cause your alternator is not charging anymore, 2003, seems about time for a new alternator, those vehicles consume lots of power cause pretty much everything is electric, lots of motors and servos, so they depend a lot on a good power source from the battery. If alternator doesnt charge it, no power...

have power system checked.

Good luck.

When you say it turns over FAST how does it sound when it turns over, or do you mean it has plenty of battery power to turn the engine over ?
EGR wires melted is not good. Must find out why they melted. Melting wires means shorting out electrical system, not good.
Do you have spark ?, fuel to the fuel rail ?