Sales growth rates at Asda and Sainsbury's are at their highest levels for three years, according to TNS Worldpanel.
Asda's 9% sales growth was the fastest of the big four, closely followed by Sainsbury's and Tesco's, which both grew by 8% for the 12 weeks to 22 April.
Total grocery market sales growth was 6%.
Tesco enjoyed the biggest increase in market share, jumping from 30.8% to 31.4% year-on-year, while Asda maintained its position in second place with a 16.8% share, up 0.5 percentage points from last year. Sainsbury's was up from 16% to 16.4%.
Morrisons' share dropped 0.1 percentage points to 11.1%, but its sales growth is the highest it has been since the Safeway takeover, at 5%.
But TNS warned that retailers would struggle to maintain strong growth as most of the market share gains from Kwik Save had already been realised.
Somerfield's share of the market dropped by 0.4 percentage points to 3.7%.
Meanwhile, Nielsen figures showed Waitrose was growing ahead of the market at 10.5% for the 12 weeks to 21 April. Sales in all the major grocery multiples rose 7.5%. "Waitrose benefited from consumers trading up over Easter," said Mike Watkins, senior manager at Nielsen.