Public support for same-sex marriage is on the rise in the latest Guardian Essential poll, and 47% of the sample say they have already voted in the postal survey.

The weekly survey of 1,841 voters finds headline support for the legalisation of same-sex marriage at 61%, compared with 58% a week ago, and opposition on 32%.

Among people who have already voted, a clear majority have cast a yes vote – 64% say they’ve returned a yes ballot and 30% a no ballot.

The Australian Bureau of Statistics reported on Tuesday that an estimated 57.5% of enrolled Australians have already cast their votes in the postal survey, with a month still to go – and just under half of the Guardian Essential survey reported they had now returned their ballots.

While 47% say they’ve already voted, 33% are reporting they will definitely vote, 6% say they will probably not vote, 3% will definitely not vote and 5% say they are not sure.

The cohort reporting they will definitely vote are split 61% in favour of marriage equality and 37% against. People saying they will probably vote are split 57% in favour and 37% against.

Only 2% of the cohort reporting they will definitely vote say they now don’t know which way they will cast their ballots, with that number 10% among people reporting they will probably vote.

The clear majority support in favour of change, and the rising headline support for a yes vote during the business end of the postal survey, will significantly boost the morale of the yes campaign – but the Equality Campaign executive director, Tiernan Brady, warned on Tuesday there was “no room for complacency”.

“The rest of this campaign we will not give up any single day, making sure that everybody out there knows how important this is – this is a vote about the lives and dignity of real people,” Brady said.

On Wednesday Malcolm Turnbull told ABC’s AM that the 57.5% turnout estimated by the ABS was a “very high” participation rate and a “ringing endorsement” of the government’s policy to give Australians a say on same-sex marriage.

The latest weekly opinion poll shows Labor has retained an election-winning lead over Turnbull’s Coalition government.

Labor is ahead on the two-party-preferred measure 54% to the government’s 46%. Last week Labor was ahead 53% to 47%. The poll’s margin of error is plus or minus 3%.

As well as tracking sentiment about the postal survey, voters were also asked questions this week about trust in institutions. The Australian federal police ranked top of a list of 15 institutions and organisations, trusted by 71% of the sample.

Religious organisations, trade unions and political parties were the least trusted organisations on the list. Trust in religious organisations has declined from 30% in October 2015 to 26% in this week’s survey.

When presented with a list of media organisations and categories, ABC television news and current affairs ranked at the top of the trust list, while commercial radio talkback programs and internet blogs were at the bottom of a list of 11.