One story that I heard was that the Cavalry used to employ a sidesaddle when they were trying to break a really rank horse because it was nearly impossible to come out of. I'm certainly not wanting to test that theory though but so far I haven't felt like I was going to come off. My mare has thrown in the odd crow hop and aside from me not appreciating her antics, I still felt secure.
Originally ladies had to ride sideways behind a man. Or sit on a saddle that had a plank on one side (to rest their feet on) and be led by a groom. These saddles were definitely not very secure at all and restricted the ladies to walking while being led. Then a saddle with an "upright head" was developed that allowed the lady to sit facing forwards. This gave her the luxury of being able to steer her own horse and go faster. These saddles were still not very secure but better then they had been. Then came the development of the "leaping head". This increased a ladies security greatly as she could now use her left leg to grip as well. Nowadays saddles with only one pommel are considered unsafe.
If your saddle fits your horse correctly and you are sitting correctly, your weight is distributed evenly. Even though your legs are off to one side.One thing riders should do to ensure their saddle is secure and thus safer is to use a girth with no elastic. The elastic can cause your saddle to not be tight enough and thus potentially slide. It's nearly impossible to find a girth that does not have elastic these days as I've discovered. I had to look high and low for one and ended up finding one at a local used tack store. New ones have to be special ordered.
The balance strap on the saddle (the one that goes from the rear right hand side of the saddle to the front left hand side of the saddle) ensures that your saddle doesn't slide sideways or "torque" when you are riding, especially jumping. See the strap that runs from the back right side of the saddle down towards the girth? That's the balance strap.
Originally ladies wore full, long skirts but they could be stepped on by the horse or caught on something, or worst of all, be caught up on the pommels of the saddle and the lady could be drug. During the edwardian and victorian times a simpler habit was developed. The lady wore tall riding boots and breeches and an "apron" overtop. This apron had much less material than a full skirt and was usually constructed so that it would come undone if the apron became hung up. Modern day riding aprons usually have a velcro closure.
Very old saddles also had a stirrup bar that resembles what most english saddles have today. These were a problem and had caused ladies to be drug if their foot was stuck in the stirrup because the leather would remain attached from the saddle. A mechanism was developed to release the stirrup leather & stirrup in the event of a fall. The mechanism remains closed when you are on the horse but if pressure is put on the mechanism (ie your weight as you come off the horse) it pops open, thus releasing everything. This is why you can't use a regular "english" stirrup leather.
Safety stirrups were also developed to prevent the foot from being caught up. These are really interesting. They basically look like a stirrup within a stirrup. The inside stirrup is hinged and has a break at the top. If you fall the stirrup basically comes apart. The inside stirrup tips backwards and then pops open to release the foot.I've also recently heard that it's a very bad idea to use a peacock stirrup (the safety stirrup with the rubber band on the outside). They are actually only meant for small children. The weight of an adults can cause the stirrup to bend.