I would think most horse riders would avoid very narrow paths, and I doubt they'd be moving fast on narrow paths, knees and trees springs to mind!
And they're used to sharing the spaces with walkers, dogs, children cyclists because they already share bridleways and roads with them, so by and large, know how to behave around other users appropriately, or we'd have reports of people being mowed down and jumped on from roads and bridleways - I'm not saying it doesn't happen but it's very rare. The same dangers exist there too, but people don't not use bridleways to walk and cycle on because of that.
As a pp mentioned, often riders pay for access to woodland, to help maintain the tracks, I used to have a permit for a local woodland, for my own and my horse's safety I stuck to the larger paths where I could see and where I could pass others safely, I could ride anywhere I wanted to within the woodland but I didn't just because I could, I also took into consideration my horse and myself and our safety, and often narrow paths have low overhanging trees that are tricky to navigate.
As for the safety aspect of women feeling safer, in 40 years experience of the 'horse world', I've never heard that as a reason to ride out.
And if you ban horses from woodland areas because of the safety aspects, then they're being pushed onto roads more - where many, many people don't want them already and aren't shy about letting you know by driving dangerously. We should be aiming to keep horses off the roads, riders for the most part don't like riding on them, but as already said here, bridleways are in decline, poorly maintained, disjoint and used by other users too.