My phone continues to restart and shut off

My phone had died and after I let it charge I turned it on and as soon as it got to the home screen it shut off and now it just continues to turn on and off. I tried taking the battery out, replacing the battery, holding down the power, home, and volume down button to clean the cache. And still nothing! The only way the phone stays on is when I put it on safe mode but when I do that the phone doesn't read my Sim card. Please PLEASE someone HELP ME!!