Moderate people did support extreme politicians. They accomplished the task by using good old fear tactics (which are especially effective, when actual rockets are flying at major cities and people get stabbed in the streets by people pronounching their will to destroy Israel)
Now, we also have a very fragmented parlament which allowed some small parties (the ultra orthodocs) to be a linchpin for coalition building. They aligned themselves with the right, because they got the funds they wanted. (The last moderate leader with a chance of being PM, Tzipi Livni, refused paying them, while Netanyahu didn't. Since then, Netanyahu crippled the Free press with a multi thronged attack. Aided by astronomical sums of money from Sheldon Adleson, created a free daily paper, which he directly controls. He focused his efforts as the minister of communications (he is actually not only the PM but the minister in a few other offices) to destroy a TV channel that was home to some very critical journalists. He managed to fragment the media to a huge amount of small players, reliant on government advertising(all in the name of making the media more democratic and free). After a very successful reform in the public broadcasting channel, done by Gilad Arden, one of his deputees, he learned that the reform didn't give him editorial power, He lamented that the reform was a mistake and tried to shutdown the public broadcasting authority. He is still working on it, he pulled his weight so no extra budget will be given to host the Eurovision competition, which we won last year (yay!)
So now the channel either exceeds it's fixed budget, and he can 'prove' that public broadcasting is inefficient. Or the channel won't host the competition, which will make all Israelis hate the public broadcasting authority.... WinWin.
Oh another example of him corrupting the israeli media was shown in the recent criminal inditment he is facing. Aledgedly he gave very preferential treatment for a Telco company in exchange for editorial control in a news website owned by the same owner of that Telco...

Using all of these new media channels under his absolute or partial control he galvanised the populus against the treacherous left, and perpetuated a very strong siege mentality (which us Jews are, understandably, prone to) while positioning himself as the only viable leader capable of fending off all those who wish to destroy Israel (which is easy, because there are plenty of people actually seeking that end)
The last few years he really upped his game by talking directly to the public through Facebook and WhatsApp, (because the media is apparently against him ️...) And using much more crude language, a la Trump. This new media operation is managed by his son.
He and his far right allies also orchestrate a public campaign against the so called left junta in the high court. This is done via his affiliated media, and by bringing about blatantly populist non 'constitutional' laws that the court rejects, which he can use as proof for the 'leftist' court.
(We don't have a formal constitution, but we have something similar)

Similarly He also promotes the idea that a leftist junta is controlling the media, the police (after his appointed chief of police didn't curtail investigations against him) and all public beaurocrats. Oh and his son is posting antisemitic memes depicting Soros copied from the American alt right. So yeah, the israeli PM is using classic antisemetic tactics against the israeli left. (We live in weird times... :( )

But, The employment rates are high, TV is entertaining and the economy is stable, so most people can easily buy in the only Netanyahu can lead us.

So if you'd ask me a few years ago, I'd say people of Israel are moderate, but fearful for the lives. Since then, the constant hammering of propoganda is corroding some basic democratic values. Younger people are less moderate, and paradoxaly are more prone to the siege mentality (although Israel is stronger than ever military wise) and nationalistic chauvinism. So I don't know if the future is bright. We still have a vocal oposition, and a moderate (albeit leaderless) majority, so I hope when he leaves the scene we'll manage to come back to some sane discourse, but I won't bet on it.