When Westwood chipped in at the 10th just after an 80-minute thunderstorm delay he was in a tie for third place.
But he took penalty drops on three of the next five holes - once from heavy rough and twice from water - in dropping five shots.
Westwood did come back with a birdie on the short 16th, but a two-over-par 73 left him seven adrift of Siem.
Winner of the World Cup with Bernhard Langer in 2006, Siem has had only one European Tour victory in approaching 300 starts.
But only a fortnight after going close on home soil at the BMW International Open back-to-back 68s put the 31-year-old in the driving seat heading into the weekend.
He double-bogeyed the 17th immediately after the resumption on play at 5pm, but then covered the front nine in a superb four-under 32 with birdies at the first, third, fifth and seventh.
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At least Westwood was still in it after his horror stretch over the same four holes that cost him four strokes on the opening day.
Open champion Darren Clarke, on the other hand, looked to have paid a huge price for one nightmare hole - a quadruple-bogey eight at the seventh, his third last hole.
That looked to have sent Clarke to yet another missed cut on four over.
The 43-year-old Ulsterman, playing his last tournament before he defends the Claret Jug at Royal Lytham in a fortnight, had been in 20th place as he stood on the tee.
But his second shot found gorse and after taking a penalty drop he went into a bunker, came out short of the green and took three more from there.
Clarke dropped another stroke on the next, but his 76 was still four better than playing partner Paul Casey.
For the former world number three that meant a second successive 80 and with only one cut made all season just like Clarke his recovery from the shoulder dislocation suffered snowboarding on Christmas Eve is taking a worrying long time.
He made his exit in last place of the 155 players.