You clearly do not know the Prime Minister or his history. He is a centrist or a moderate. Always has been, always was, even back when he was with the Liberal Party (merged into Likud) or his own Shlomtzion party.
Also, don't assume American political terminology (i.e.: that forward somehow equals progressive) applies in Hebrew. The only reason for the name change is that Aharayut Leumit was a mouthful.
But he embraces the Leftist policies of the Marxist Labor party. Don't get me wrong, Amir Peretz the Union leader as Prime Minister would be 1-second National Suicide, but Sharon has always been a leftist. Like Hillary Arafat, he is a skilled politician so he can make it so he looks like a right winger. The Israeli version of Tom Tancredo is Itmar ben Gvir who is a leader of the Chayil (Valor) party. THAT is someone I want in the Knesset. Chayil is growing in support particularly the youth, and if it gains seats in the Knesset we could see some real Right-Wingers gain power.
Well, I have some familiarity with Jewish involvement in progressive and revolutionary causes, and the word "forward" does tend to have progressive implications for Jews as well as others.
Many Jews still consider themselves to be progressive, New York and Miami Jews being obvious examples.
I'm familiar with the Jewish Daily Forward mostly because of Isaac Bashevis Singer, whose work I greatly admire, though not his politics.
I know Sharon was a military hero, and I know he belonged to the more conservative of the two main parties. But he seems to be one of those rare people who move left when they get older instead of getting more conservative. It's hard to be sure of that, but that's how I read his behavior since he assumed office this last time.