The SNP picked up key seats in Edinburgh Central - where former SNP Westminster leader Angus Robertson replaced the one time Scottish Tory boss Ruth Davidson - as well as as in Ayr and East Lothian.
But under Holyrood’s proportional representation system, those successes could see it lose seats on the regional list ballot.
Meanwhile, Labour’s Jackie Baillie held on to her Dumbarton constituency - which had been the most marginal seat in all of Scotland and a top target for the SNP.
Ms Baillie had a majority of just 109 from the 2016 Scottish Parliament election, but increased that 1,483.
With some constituencies still to be counted on Saturday, when the crucial regional list results will also be declared, Ms Sturgeon said it was “not impossible”.
With 47 constituency results declared on Friday, the SNP had 38 seats, Liberal Democrats four, Conservatives three and Labour two.
The coronavirus pandemic meant traditional overnight counts were abandoned after Thursday’s Scottish Parliament election.
And while the majority of the 129 MSPs at Holyrood have still be declared, Ms Sturgeon said it was “almost certain” the SNP would win its fourth term in power at Holyrood.
She also stressed that “when the time is right”, she should be able to offer Scots “the choice of a better future” in a second independence referendum.
Ms Sturgeon, who comfortably defeated Scottish Labour leader Anas Sarwar to claim Glasgow Southside, said afterwards: “My focus, if we are re-elected as the government, is to get back to work to steer the country through the crisis and into recovery.
“That remains the case. But once the crisis is over, and if there is a majority in the parliament for an independence referendum, people should have the right to choose our future. Scotland’s future should always be in Scotland’s hands.”
Speaking about the prospect of winning an overall majority, the SNP leader said: “It’s certainly not impossible, but nor is it guaranteed.
“That was always going to be on a knife edge, it comes down to a small number of votes in a small number of seats, so at this midway point it is certainly still there as a possibility, but I have never taken that for granted.
“It is a long shot, to say the least, in a PR system, to win a majority, you effectively have to break the system. I would like to do it, but I have never been complacent about that.”
However she said it was “almost certain” the SNP would “win the election comfortably, and we should not understate the scale of that achievement”.
Meanwhile Mr Robertson, the new Edinburgh Central MSP, said the message from voters there was that “Scotland’s future should be in Scotland’s hands”.
The former SNP depute leader insisted: “In this most European of capital cities, people have resoundingly rejected the party of Brexit and Boris Johnson.
“The public has rejected all of the parties that want to block an independence referendum.”
As he secured his Perthshire North seat, Deputy First Minister John Swinney said the SNP would be the “leading and largest party” in the new Scottish Parliament.
While he said there is a “long way to go” before all the results are known, he stressed it was now “beyond any doubt” that the SNP will form the next government.
He added: “That is an absolutely gigantic feat for the Scottish National Party to have achieved, to be on the brink of a fourth continuous term.”
Elsewhere, former first minister and Alba Party leader Alex Salmond said the measure of his party’s success would be “our existence as a political party”, adding it is “here to stay”.
Prime Minister Boris Johnson again expressed his resistance to the idea of another independence referendum, saying it would be “irresponsible and reckless” in the “current context”.
Pressed on what he would do if Ms Sturgeon pushed ahead with a referendum without Westminster’s consent, he told the Daily Telegraph: “Well, as I say, I think that there’s no case now for such a thing ... I don’t think it’s what the times call for at all.”
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