Prime Minister Benyamin Netanyahu has large followings of those who love him, and those who hate him.
The love side of Bibi's audience appears in this recent note from an Internet friend, which resembles what I have seen from others.
Regarding Bibi, I think he walks on water . . . . In studying his life and accomplishments, he is really, one of a kind. I believe that he has only the true values of Israel in his heart and mind, 99% of the time, but being imperfect, he too will make some mistakes. However, he usually, when these are called out, will try to rectify them and apologize. He is the world's greatest defender of Israel and its people, and every time I see another video of him before the IDF, as yesterday, right on the border with the Golan Heights, I just want to get up and hug the man. He is so brilliantly clever, devoted to his job and country/people, with such intelligence, command of languages, customs, history, and always defending Israel, no matter who is on the other side of the fence (like Obama/Kerry and their friggin, stupid deal with Iran- how stupid!). . . . How can there be so very many, American Jews who are on the fence or so far to the left that they do not understand or support Israel? 
For the other extreme, we can look at the scatological Facebook entry of Simone Zimmerman, formerly associated with the presidential campaign of Bernie Sanders. 
" . . . arrogant, deceptive, cynical, manipulative asshole . . . tone-deaf . . . towards the international community . . . Fuck you, Bibi, for daring to insist that you legitimately represent even a fraction of the Jews in this world, for your consistent fear mongering . .  . trying to derail a potentially historic diplomatic deal with Iran and thus trying to distract the world from the fact that (you) sanctioned the murder of over 2,000 people . . . a brutal military occupation of millions more . . . "
Chronic Israeli criticism is less salacious, but focused on assertions that Natanyahu serves the interests of big business, provides only slogans for the sake of the country's poor, has maneuvered the country into a hostile corner of international opinion, and has done nothing to advance whatever chances there are for reaching an accord with the Palestinians.
Bibi's supporters consider his domestic and international critics blind to realities. Their latest target is US Vice President Joe Biden, who expressed "overwhelming frustration" with Israel's government, as well as 

". . . the actions that Israel’s government has taken over the past several years—the steady and systematic expansion of settlements, the legalization of outposts, land seizures—they’re moving us, and more importantly they’re moving Israel, in the wrong direction . . .  one-state reality . . .(that) is dangerous."

Netanyahu has been part of Israel's political elite since serving as Ambassador to the United Nations in 1984. He first served as Prime Minister 1996-99, later served as Finance Minister and Foreign Minister under Ariel Sharon, and again Prime Minister from 2009 to the present. He's close to David Ben-Gurion's record as the longest-serving Prime Minister, 
​Bibi's present coalition has an unenviable margin of one vote in the Knesset, and he is beset with the fatigue of voters and party colleagues directed at someone who has served so long, and taken decisions that offend some while they please others. 
Bibi's most prominent nemesis among Likud Knesset Members is Oran Hazan, a former casino manager in Bulgaria, alleged to have served as a procurer of prostitutes and hard drugs. Hazan has ridiculed a handicapped Knesset Member, and urging the destruction of Muslim holy sites to allow the construction of a Third Temple.
Bibi has lost government-supported Knesset votes due to the rebellious violations of party discipline by Hazan and others. The Prime Minister has also reached the level of political threats (shut up or I'll fire you) in exchanges with coalition colleague Naftali Bennett of Jewish Home. 
However, it's a long way from not being able to carry every vote defined as a government measure to actually losing his position as Prime Minister. That requires a majority vote by the Knesset to either call for an election, or to name another Prime Minister. Although Bibi has problems with his own party and other coalition partners, the political arithmetic is stacked against the probability of replacing him.
Sara doesn't help. In a setting where a politician's family is generally out of the public eye, Sara has an exceptional record that is almost totally negative. She's known for yelling at the household help, extravagance with public resources, pressing her likes and dislikes on Bibi with respect to the appointment and dismissal of key government officials, bizarre demands for food or drink, and one notable tirade against Bibi.
With all that, there is no evidence that Sara's behavior has done more than entertain media watchers. Bibi has supported her publicly in a number gaffes, yet remains in office. If he's vulnerable politically, it is on account of his own actions and not Sara's.
Bibi's opponents and supporters say or imply that he is moderate in action if not in words. Opponents to the right decry this as timidity, and failing to take actions that firm up Israel's security. His supporters note that he has sought to avoid direct public confrontations with foreign leaders, including Barack Obama, even while opposing policies he sees as threatening Israel. He has ruled against religious activists in his coalition in order to lessen tensions with Muslims with respect to the Temple Mount, and has stood against political allies who want to expand settlements in the West Bank or pursue the conquest of Gaza.
Bibi has something to do with the antipathy toward Israel among American Jews. He has been the head of the government whose policies critics attack. His defense of the status quo does not go over well with young people cut off from Israeli realities. 
It's hard for an Israeli to identify with those who say that Netanyahu is a barrier to peace, given the bloody absence of peace all around us, and the Palestinians' rejections of several opportunities that can be described as reasonable or forthcoming.
The American Vice President may be unhappy with Netanyahu, perhaps especially when speaking with JStreet. Closer to home, however, where the votes are, the leader of Israel's opposition has sought to position his party closer to the Netanyahu constituency, with a comment that the Labor Party should avoid giving the impression that it loves Arabs.
There are many Israelis who have tired of Bibi's blather, and what seem like pointless quarrels with prominent outsiders.
However, the ultimate judge of Netanyahu's success is the measure of any politician's success, i.e., staying in office. 
By that criteria he has done well, and has no obvious replacement poised for a take-over.
Comments welcome
Ira Sharkansky (Emeritus)
Department of Political Science
Hebrew University of Jerusalem