Clanton and Suwannapura birdied five of the last seven holes on the front nine and played the last five holes in 2 under, making three birdies and a bogey.
"I think because we pretty chill, like 'You hit a bad shot, like OK, I'll fix it, don't worry,'" Suwannapura said. "So everybody kind of like relaxing the game today, so the result pretty good."
They had a 16-under 194 total at Midland Country Club, with a best-ball round left Saturday.
"Each hole's an opportunity," Clanton said. "How many opportunities can we give ourselves. One of those things where as many fairways and greens as you can hit and if the putts go in, great."
Suwannapura, from Thailand, won the Marathon Classic last year for her lone LPGA Tour title. Clanton is winless on the tour. They had an alternate-shot 67 on Wednesday and shot a best-ball 64 on Thursday.
"Today, I think what we did was quite impressive just with our ball-striking in general," Clanton said. "Jasmine's putting's on point. I think you've just got to look at it as it's just another opportunity to play and the more that you can think of the fact that it's just another day on the golf course, we're going to go out and try to do what we've been doing, the better it's going to be."
Sisters Ariya and Moriya Jutanugarn had a 64 to move into a tie for second at 11 under with Simin Feng-Ruixin Liu (67), Na Yeon Choi-Jenny Shin (67) and Paula Creamer-Morgan Pressel (69).
"I felt like first round, don't want to miss any shot because I don't want my sister to have like tough shot and I kind of worry and scared," Ariya Jutanugarn said. "But today we already make the cut and nothing to lose, so we keep playing golf."
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