star wars - Why Did Anakin Skywalker Believe that Turning to the Dark Side of the Force Would Save Padmé?

Just as the question asks: Why did Anakin come to believe that Padmé could be saved if he embraced the Dark side of the Force? Padmé clearly didn't believe it. Why was Anakin so driven to save Padmé through the dark side?

Answer

When he first looked for advice from Yoda about his vision, he did not hear what he wanted to hear. From Star Wars Episode 3 Revenge of the Sith Script

YODA: Careful you must be when sensing the future, Anakin. The fear of loss is a path to the dark side.

ANAKIN: I won't let these visions come true, Master Yoda.
YODA: Death is a natural part of life. Rejoice for those around you who transform into the Force. Mourn them, do not. Miss them, do not. Attachment leads to jealousy. The shadow of greed, that is.
ANAKIN: What must I do, Master Yoda?
YODA: Train yourself to let go of everything you fear to lose.

But later, Palpatine told him about a possible way to save her :

PALPATINE: I thought not. It's not a story the Jedi would tell you. It's a Sith legend. Darth Plagueis was a Dark Lord of the Sith, so powerful and so wise he could use the Force to influence the midi-chlorians to create life ... He had such a knowledge of the dark side that he could even keep the ones he cared about from dying.
ANAKIN: He could actually save people from death?
PALPATINE: The dark side of the Force is a pathway to many abilities some consider to be unnatural.

So, Anakin's dilemma was that Yoda told him he must accept Padmé's death and Palpatine told him he could avoid it by turning to the Dark Side.

Palpatine also told him that the Dark Side is not Evil as the Jedi told him and made him doubt the real intention of the Jedi order, that they where hanging on power and that their actions where not always for the greater good. In the end, Anakin turned to the Dark Side to save Padmé, but also because he had some conflict of opinion with the order. He was convinced that he was doing the right thing.