I think the general consensus is it's not normal to feel your pelvic bones shift with each pedal.
That makes your sitch sound like the seat is be too high.
How many miles/hours have you spent in the saddle? How often have you ridden? How long have you been riding? Before I started using a trainer in the winter, every Spring I had to treat my butt very gingerly - a couple of miles one day, a day or 2 of rest, then 4 miles, and a day or 2 of rest, and so on. So if you're talking about just starting out, and perhaps forcing yourself to ride consecutive days, what you describe may be normal, if it's not the seat being to high.
Also, leather saddles conform to the rider. If you see indentations where the previous owner's sitbones might have been, you might be riding a saddle that just won't fit.