Only one in five trains were running on Saturday as more than 40,000 members of the Rail, Maritime and Transport (RMT) union walked out in a long-running dispute.
Mr Lynch accused the Conservative government of tabling unrealistic proposals in negotiations aimed at resolving a row over pay and working conditions.
Despite the lack of progress in the months-long discussions, Mr Lynch vowed his union's members would stand their ground.
“If we gave up we would lose everything that we’ve negotiated over the decades and we’re not prepared to do that,” he said, speaking from a picket line outside Euston Station in central London. “We will be in this for as long as it takes.”
Mr Lynch, who was criticised by Liz Truss as being part of an “anti-growth coalition”, accused the prime minister of mangling the economy in the midst of a cost-of-living crisis. He also said he had no confidence in Chancellor Kwasi Kwarteng.
“She seems to be an incompetent — her incompetence is only matched by her ego, and the same can be said for Kwasi Kwarteng,” he said.
“They’re saying that we’re an anti-growth coalition, they had to go out and print £65 billion to prop up the bond market. But they want to blame railway workers for what’s going on in the country.”
About half of the network will be closed all day on Saturday and trains are operating until 6.30pm in continuation of the disruption that occurred during the summer strikes.
Passengers have been urged to “only travel by train if absolutely necessary”, with services expected to return later than normal on Sunday.
Thousands of replacement staff have been drafted in to limit the effect of the strike, but only a fifth of normal services are running.
There was hope of a breakthrough in talks between union bosses and government officials when Ms Truss appointed Anne-Marie Trevelyan as transport secretary in September, replacing Grant Shapps. Ms Trevelyan said was a “deal to be done” with the unions.
In a speech to the Conservative Party conference in Birmingham last week she called for “compromise” and “positive proposals” to end the upheaval to nationwide travel.
Mr Lynch said Ms Trevelyan had “warmer words” than her predecessor but had failed to bring concrete proposals to the negotiating table.
“We know that there’s a deal to be done but it’s whether the government allows these (rail) companies to change their position,” he said. “At the minute, they’re saying things that they know that we can’t accept.
“The companies know that we can’t accept — they say to me, we know you’re not going to accept this, but I’ve been told to say it.”
The Rail Delivery Group, which represents the rail companies, declined to comment in response to Mr Lynch's statement.
The Department for Transport said in a statement: “For the third time this week, and second weekend in a row, we face disruption and disarray on our railways.
“Weekend rail travel has been on the rise since the pandemic as people travel to support their favourite sports teams, meet loved ones or go to events. These plans are all now jeopardised and our economy is damaged as a result.
“Our railway is in desperate need of modernisation but all strikes will do is punish the very people unions claim to stand up for and push passengers further away.
“We urge union bosses to reconsider this divisive action and instead work with their employers, not against them, to agree a new way forward.”
Three strikes have taken place so far in October, with the others being held on Saturday 1 and Wednesday 5.