Bullfinch

From the ubiquitous Wikipedia:

This fence has a solid base with several feet of brush protruding out of the top of the jump up to six feet high. The horse is supposed to jump through the brush, rather than over it. Due to the height of the brush, the horse generally cannot see the landing. This tests the horse's trust in the rider, as the horse must depend on the rider to guide them carefully and steer them to a solid landing. The horse must be taught to jump calmly through the brush, as their attempting to jump over the brush could lead to a refusal, a run-out at the next fence, or a misstep and possible injury. Bullfinches must be approached positively, with lots of impulsion, in order to prevent stops. When jumping a bullfinch, the rider must stay tight in the saddle so that brush cannot be caught between their leg and the horse.

From the Duke of Beaufort in his book, Fox-Hunting:

Bullfinch.  A hedge so high that you cannot jump it, but not so thick that you cannot jump through it.

From Margaret Cabell Self in her book, The Horseman's Almanac and Handbook:

Bullfinch.  This is a type of jump found in England.  It consists of a thick hedge, often of thorn, too high to be jumped and through which the rider must bore.  There are often ditches and guard rails on one or both sides.  The bullfinch is terrifying to one not used to it and to horses from foreign parts, but it is very common in certain parts of England.  A black bullfinch is one so thick that one cannot even see through it.

From John Welcome in his book, Red Coats Galloping:

. . . "God Almighty, it's Jeremy!  What in the name of all that's wonderful are you doing here?"

There were several replies which Jeremy might have made but he chose none of them, for the simple reason that he was much too preoccupied with the problem of dealing with the fence ahead.  It was another fence of which Jeremy had read much but had never before seen -- a bullfinch.  The huntsman had gone through it higher up and there was nothing for it but to harden his heart and ride on.  He put down his head, hunched up his shoulders and sent the mare at it.  With never a moment's hesitation she took him through that big black fence with the power and velocity of a sixteen-inch shell. . . .

 You might hunt your whole life throughout North America and never see a Bullfinch, let alone jump one.   [Thank God!]