People keep on saying this. It isn't true. You might consider what happened to Stephen a few years later. You might consider that at one point in His ministry they were ready to stone Him and at another to throw Him off a cliff.
There was another reason why they handed Him over to the Romans.
I know that they stoned Stephen, but they took Jesus to Pilate because they did not have the authority to put someone to death.
John 18:28-31 - Then led they Jesus from Caiaphas unto the hall of judgment: and it was early, and they themselves went not into the judgment hall lest they should be defiled; but that they might eat the passover. Pilate then went out unto them and said, What accusation bring ye against this man? They answered and said until him, If he were no a malefactor, we would not have delivered him up unto the. Then said Pilate unto them, Take ye and judge him according to your law. The Jews therefore said unto him, It is not lawful for us to put any man to death.