As I've already said. A petty burglary doesn't equate to committing 5 murders.
Jeremy got it in first before the mis-trusting girlfriend did. He wasn't obliged to admit to doing it,,the same as he wasn't obliged to have done anything other than turn over after his father rang him on the night of the murders. In fact,,in view of what happened to him,,he may as well have stayed in bed,because nobody would have been any the wiser considering that there was no forensic evidence to link him with any murders. I wonder what the police would have done then ?? Apart from scratch their heads and behinds.