The 34-year-old Indian, who is hoping to work his way to getting a full card on the European Tour once again, had six birdies against two bogeys in his card of four-under 68.
The Nordea Masters is part of The Open Qualifying Series, with the leading player, not otherwise exempt, who finishes in the top five and ties this week will qualify for The 2016 Open Championship.
On the back nine of the Bro Hoff Slott Golf Club, Kapur dropped a shot on 11th to come down to one-under for the day and one-over for the tournament. At that stage he was still in the danger zone of missing the cut.
Kapur fought back well with birdies on par-5 12th, par-3 16th and par-5 18th to finish at 68 and move to Tied-34th, an upward move of more than 65 places. But many of the golfers were yet to finish their second round.
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Matthew Fitzpatrick signaled his fine form with a stunning round of seven-under 65 that gave him a 36-hole total of 11-under 133 and a one-shot lead over Belgian Nicolas Colsaerts (70-66) and two shot advantage over Frenchman Thomas Linard (69-68).
Yesterday, Kapur and Jeev had a disappointing start to their campaign. Kapur shot two-over 74 while Jeev carded three-over 75. Daniel Chopra carded four-over 76.
Kapur was going fine at two-under through the front nine, before finding four bogeys on the back stretch while Jeev, starting from the tenth, was even par at the turn with two birdies and two bogeys. Then he had three bogeys on the front stretch of the course for his 75.