When G̶o̶d̶z̶i̶l̶l̶a̶ a Zillo Beast is unearthed on Malastare in the second season of The Clone Wars Chancellor Palpatine makes the decision to capture it and use it for scientific research. Palpatine sees the possible military benefits of keeping the beast alive.
DOCTOR BOLE: Chancellor Palpatine, I think it would be in our best interest to study this creature. Its outer scales are virtually impenetrable. None of our weapons could harm it.
ANAKIN: Even my lightsaber couldn’t cut it.
DOCTOR BOLE: If we could somehow duplicate that for our ships...
PALPATINE: An impenetrable armour.
ANAKIN: What if we let the Dugs think we killed it? Then we could have the treaty and the creature.
PALPATINE: Yes, Anakin, go on.
ANAKIN: When I was climbing on the creature I noticed small gaps between the armour and its scales. Maybe a stun cannon or a pulse could penetrate through the skin and short-circuit its nervous system. Like putting it into a deep sleep.
DOCTOR BOLE: He’s right. It would appear to be dead. We could transport it to a secure location before it wakes up. The Dugs would never know the difference.
PALPATINE: That may be the only way we shall get this treaty signed and rest Master Windu’s conscience.
(The Clone Wars, Season 2, Episode 18, "The Zillo Beast").
Note that it is Palpatine's personal decision to capture the Zillo Beast (to keep Windu happy) rather than kill it. He also personally oversees the choice of planet on which to test it.
WINDU: The beast has been loaded onto the transport. Have you selected a planet for relocation?
PALPATINE: There has been a slight change of plans. The scientific community has reviewed the data on this rather unique creature. They believe it should be studied in a more controlled environment before it is released into the wild.
WINDU: And where is this controlled environment?
PALPATINE: On Coruscant.
(The Clone Wars, Season 2, Episode 18, "The Zillo Beast").
Predictively enough, perhaps, everything goes horribly wrong. The Zillo Beast, enraged by the experiments being performed upon it, escapes and proceeds to wreak havoc upon Coruscant.
Bear in mind that it is described as "the most dangerous life-form in the galaxy".
NARRATOR: Tensions run high as the most dangerous life-form in the galaxy touches down on the Republic’s most populous planet.
(The Clone Wars, Season 2, Episode 19 "The E̶m̶p̶i̶r̶e̶Zillo Beast Strikes Back").
Here it is, destroying Republic ships...
...stamping on bridges, killing lots of people...
...setting fire to parts of the planet...
...and even attacking the Senate building.
This last point is the most important one I think. Palpatine needed to maintain his popularity in the Senate in order to execute his wider plan of establishing the Empire. If he'd been kicked out then he would never have become Emperor. Yet think about the damage the Zillo Beast does here. It kills and injures untold numbers of people (probably thousands). It destroys large parts of the city. It attacks the Senate building itself, the workplace of all the Senators - thereby putting their safety at risk. Even if Senators hadn't felt concerned for all the citizens of Coruscant who'd just died surely they would have felt threatened themselves by the Zillo Beast - and wanted to hold Palpatine responsible.
We have the parallel of Supreme Chancellor Valorum being ousted from office in Episode I for failing to respond to an invasion on a far-flung planet. How did Palpatine's political career survive the attack of a huge monster that destroys all in its wake? On the roof of the Senate of all places. After all, he was personally responsible for the decision to bring the Zillo Beast to Coruscant in the first place.
Is there any evidence in canon that Palpatine's reputation in the Senate took a hit from this incident? Why wasn't he impeached in a similar manner to Valorum?