Goldsmith was arguing with Trevor Mallard, pushing back strongly on his decision not to let him continue his line of questioning of Education Minister Chris Hipkins in the House.
After a few minutes of back and forth, Mallard told Parliament it was time to move on.
During the exchange Mallard told Goldsmith he needed to listen "by using your ears", to much laughter in the House.
Goldsmith could be heard saying "waste of time" as he was leaving the chamber.
Bridges called Mallard a "twat" as he was leaving, according to an MP to heard the exchange.
Goldsmith, who said he still had one question left, stood up and stormed out - yelling something at Mallard as he left.
Bridges, who was not involved in the argument, then followed his fellow MP out.
National has been at odds with Mallard this year, including attempting to move a no-confidence motion with the Speaker.
It is understood that shadow leader of the House Chris Bishop will attempt to move the motion again after Question Time today.
But - like last week - the motion is doomed to fail as Labour has a majority in the House.
National want the Speaker gone, after it was revealed last year that Mallard cost $300,000 of taxpayers' money to settle a defamation claim, after he falsely claimed an accused rapist was working at Parliament.
More than a year later, Mallard apologised to the staff member in question, through his lawyers, in a public written apology.
Although any no confidence vote is doomed to fail, as Labour holds a majority in the House, Bishop had said that this was a "matter of principle" for National.
"We're very clear that his behaviour is not up to the job of the Speaker," he said earlier this month.
"It's just simply not appropriate to have the Speaker of the House besmirching the dignity of Parliament in the way that he has and failing to uphold the standards of the House that he himself is in charge of enforcing."
It's likely National will continue to lodge no confidence motions in the coming weeks and months.
"We have about 200 sitting days this year, which I'm sure that the Speaker will have notices of motion in there for many of those," National leader Judith Collins said earlier this month.
Collins said the no-confidence vote will be lodged "on the basis that we think we should".
Asked if National planned to continue moving the motion over the coming weeks, she said: "Hopefully, it won't be needed. Hopefully, the Prime Minister will take this matter seriously."
But Jacinda Ardern had previously said that Labour would continue to support Mallard in his role as Speaker.
In December last year, she admitted that Mallard had "made a mistake".
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Comments
Mallard must be the most biased speaker in living memory. He really is unfit for the post.
David, you ae just saying that because you want to because you wear blue or based on media reports.
I watch Parliament TV frequently and have for years.
Mallard would be one of the best and most unbiased.
He operates a tight pedantic ship.
He also has an open door policy for any MP to meet privately over matters.
Just today he addressed the House about matters to do with questions.
I heard the debacle today and Goldsmith was out of order.
An MP is supposed to keep an eye on the Speaker at all times and address him.
When the Speaker stands up, you sit down.
Goldsmith didn't and carried on talking - claiming he never saw the Speaker stand up or talk.
Stick to unbiased facts please David.
The first thing I note is that this article is from the NZ Herald. That fact immediately prejudices my reading in that I take anything contained therein with a grain of salt. GM's comments above, as someone who heard the event as it happened has a completely different take on proceedings.
I too regularly watch parliament on TV and I concur with GM, Mallard is one of the best Speakers we have had. He is certainly a significant improvement on the previous incumbent, David Carter. Carter had to be the most incompetent and biased Speaker in my memory, especially compared to Lockwood Smith and Margaret Wilson before him.
Doesn't mean that Mallard hasn't made some foolish mistakes outside of the debating chamber but his chairmanship of parliament is well above par.
It would seem that par is a very low score
Par is the result expected from a competent or even expert participant. If you're achieving par you're doing extremely well. I said that Mallard was well above par, meaning he he was doing better than par, of course because of the perversity of the English language I should have said he was under par, he's an eagle!