Thousands of people have marched in Glasgow in the largest show of support for Scottish independence since Nicola Sturgeon said she would introduce legislation to hold a second referendum on the issue.

The All Under One Banner event, led by a single flag-bearer and a pipe band, left Kelvingrove Park at 1.30pm and was following a route west to east through the city centre to a rally at Glasgow Green.

Marchers were still streaming out of the park at 2pm, many carrying Scottish and EU flags. There was a carnival atmosphere as families and dog-walkers joined drummers and bagpipers, and there were chants of “independence now”.

Marchers still streaming up Gibson Street onto Woodlands Road for the #AUOBGlasgow event pic.twitter.com/fsjo7rjP6t — Libby Brooks (@libby_brooks) May 4, 2019

Police Scotland said it was difficult to estimate numbers accurately because people had joined the march at different points along the route, but by 5pm they believed there were between 30,000 and 35,000 people gathered at the rally on Glasgow Green.

Sturgeon, Scotland’s first minister, said at the end of April that a second independence referendum should be held by May 2021 if Brexit went ahead, but stopped short of repeating previous calls for the UK government to give her the power to call one.

All Under One Banner, which describes itself as “open to everyone who desires an independent Scotland”, has marches planned across Scotland throughout the summer, culminating in an event in Edinburgh on 5 October.

Writing in the National on Friday, organisers alerted attendees to the possible presence of counter-demonstrations along the route. “Marchers can either ignore the counter-demonstrators or smile and wave … our movement doesn’t need any negative publicity,” they said.

Traffic Scotland advised drivers to plan ahead as access to the M8 motorway would be shut off through the city centre.

The march is one of a series of events taking place across Scotland between May and October. Photograph: Jeff J Mitchell/Getty Images

Meanwhile, SNP activists were distributing half a million “Stop Brexit” flyers across the country on the first of three campaign Saturdays in the run-up to the European elections.

The party’s spring conference, held in Edinburgh last weekend, called on voters to use the EU elections as an opportunity to reinforce Scotland’s support for remaining in the EU.

Two polls, by YouGov and Panelbase, have suggested the SNP will win three or four of Scotland’s six European seats, reducing Labour and the Tories to one each. YouGov showed the SNP on course for 40% of the vote.

Sturgeon told the conference: “As voters go to the polls for the European elections on May 23, our message will be clear and direct. And unlike Labour’s, it will be unambiguous. Scotland’s not for Brexit, Scotland’s for Europe.”