In Revenge of the Sith Palpatine says he wants all the Jedi to be killed in his speech and the Republic cheers. What I am wondering is what made them turn on the Jedi? Didn't they know the Jedi were protecting them? They must have seen some value in them if they were giving them missions still. What made them loose trust in the Jedi all of a sudden?
They were under the influence of the Sith
A quote from Star Wars: Attack of the Clones, during Obi-Wan's capture by Count Dooku:
Dooku: The truth. What if I told you that the Republic is now under the control of the Dark Lords of the Sith?
Obi-Wan: No, that's not possible! The Jedi would be aware it!
Dooku: The Dark Side of the Force has clouded their vision, my friend. Hundreds of senators are now under the influence of a Sith Lord called Darth Sidious.
So they were already under Sith influence, via Palpatine's skilled words manipulation at the least (and at every step insisting they give him their complete trust and more power), and probably some degree of force power clouding their better judgement. Add this to Palpatine setting the Jedi up to look like they attempted an assassination, and it was an easy thing for the Senate to buy. Why doubt the poor weak Supreme Chancellor, over those war-mongering Jedi who probably secretly became corrupt and wanted to take over?
Palpatine told them the Jedi planned to overthrow the Senate
The reason they cheer and applaud Palpatine's plans to seek out and destroy the Jedi was because he had led them to believe the Jedi were planning to overthrow the Senate. Palpatine used his prior confrontation with Mace Windu as evidence that the Jedi attempted to assassinate him - fairly convincing evidence, I'd assume, given his disfigured face.
The gullible Senate fell for it hook, line, and sinker.
The Chancellor is in the podium in the center of the vast arena giving a speech. MAS AMEDDA stands to the right of SIDIOUS. BAIL ORGANA walks through the hallway of the Main Senate Chamber. He enters the Senate Pod of Naboo and sits next to PADME. JAR JAR, CAPTAIN TYPHO and TWO HANDMAIDENS are in the pod also.
PALPATINE: . . . and the Jedi Rebellion has been foiled.
BAIL ORGANA: I was held up. What's happening?
PADME: The Chancellor has been elaborating on a plot by the Jedi, to overthrow the Senate.
BAIL ORGANA: That's not true!
PADME: He's been presenting evidence all afternoon.
BAIL ORGANA: And the Senate will go along with it, just like they always do.
PALPATINE: The remaining Jedi will be hunted down and defeated. (applause) Any collaborators will suffer the same fate.
These have been trying times, but we have passed the test.
PALPATINE: The attempt on my life has left me scarred and deformed, but I assure you my resolve has never been stronger.
Jedi were defenders of corrupt Republic for too long
At the beginning of Phantom Menace, Republic is horribly inefficient and decadent beast. Senate cannot even decide is one of the member planets invaded ! Not to mention representatives from large corporations in legislature, teenage Senators, bureaucracy etc... And outside of Senate, extreme wealth of monarchs on on side (Alderaan) and extreme poverty on other side, especially in Outer Rim. All of this could not be attributed to workings of couple of Sith Lords, simply saying roots of disaster were inside very core of supposed democracy.
What about Jedi ? Jedi are secretive Force sect, rumored to "voluntarily" take children from their families, with supernatural powers. Yet, they act like ultimate police force and enforces of Republic, with little control by the public. Or simply, they are the fist of current regime.
Clone War starts, Jedi suddenly become generals of massive Clone Army. Note that other citizens of Republic mostly do not participate in war, they are unarmed victims and thoroughly depend on others to protect them. There were exceptions, but what most of the people see is Jedi and clones on "our" side, and former Jedi and droids on "their" side. Because, Dooku is a well-known former Jedi, and probably many wondered is he really "former".
Role of Palpatine. While we know that he is a Sith Lord, general public would see him in a positive light. He was less corrupt then other politicians, remained humble, and was practically only "ordinary" being controlling vast army of Force-sensitives and clones. Basically, he was someone general public could relate to. Jedi try to overthrow him, details are unknown but is certainly illegal. They fail and are removed, and suddenly Clone War with all devastation stops.
No Jedi no problem ? It certainly looks that way. One of Jedi is leader of CIS, other Jedi attempt a coup, all of them gone and war is gone. Was war just a plot by Jedi to take over ? It looks plausible. Anyway, Palpatine controls Senate, declares Empire, general public is happy to see Old Republic gone and buried, along with Jedi Order that was symbiotically attached to it.