In Thursday's poll, which also included elections to the Scottish and Welsh assemblies, Labour held its ground in England but suffered a bruising night in Scotland.
In a debate on Thursday about the significance of the results for the upcoming referendum on 23 June, Denis MacShane, a former Europe Minister in the UK, clashed angrily with Matthew Ellery, of the Get Britain Out campaign group.
Ellery disputed MacShane's assertion that Eurosceptics were isolationist, saying it was wrong to "slur" Eurosceptics as such.
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He said, "Nothing could be further from the truth. We are internationalist, not isolationist. We believe the UK shouldn't focus itself solely on Europe; we should cast our eyes to broader horizons and engage with the world."
Ellery also noted MacShane's "failure to support the democratic decision being made in the UK local elections, something which cannot be achieved in the EU, with the European Commission being unelected."
Ellery described the local elections as a "fine display of democracy in action, with the people of this country voting for those who govern them."
In the online debate, he added, "Unfortunately, the same cannot be said of the European Union, with executive power vested in 28 unelected Commissioners."
He said the "widespread appeal of Brexit" was that people from "all types of backgrounds" had the chance to "support democracy by urging the country to leave the EU."
Ellery, whose group represents grassroots public opinion in the UK, also disputed that a Remain vote is not looking more likely.
He said, "The polls, for what they are worth, are saying the vote is neck and neck, with the Leave campaign in the ascendancy, especially with those who are certain to vote. This is because those opposing the European Union are very passionate in defending the fundamental right of democracy and will do everything in their power to regain the ability to hold those who govern them to account."
But MacShane, a cabinet minister under Tony Blair, hit back by defending his assertion that Eurosceptics were isolationist.
The former Labour MP also believes that outgoing London Mayor Boris Johnson will not have "as big a splash" on the Leave campaign as he expected.
He said, "His Donald Trump big mouth style may not be suited to a long campaign where calm exposition of factual arguments may have more impact. Nigel Farage with his anti-immigrant passions is making an impact as well as top Tories like Ian Duncan Smith, Michael Gove and other isolationist Conservatives."
MacShane said, "Over the weekend we will absorb the results of Super Thursday and then new energy will have to be found to knock on doors and hand out leaflets for the EU plebiscite."
He added, "Something to watch for is turn-out in the local elections. I sense from my own canvassing for Labour that there was little excitement in the election. If there is a low turn-out that may suggest a same lack of participation on 23 June in the referendum. That is very good news for Leave as a small turn-out will benefit the pro-Brexit camp."
He also believes a Brexit would hit the UK economy, saying, "I would say that on the whole most authoritative economic sources, from IMF to OECD, and most serious business leaders are very concerned that Brexit would cause considerable economic dislocation.
"The loss of the single EU trading passport for hundreds of City banks and financial firms would be a worry as outside the EU they would have to relocate part of their business to an EU member state and the right the UK has to control most trades and clearing in euros would disappear once the UK was not in the EU."