Bloody from the smooth part of the bar is not a good sign. Was it swinging to your shins or just from friction on the drag up? If swinging in you need to set up closer to the bar. If just from friction then it may have been incorrect initiation of the movement. If you start with the shins at a slightly forward angle and initiate the movement with a hip extension your knees will shoot forward and into the bar path. If you start the movement with your back anchored and initiate with a knee extension your shins will go more vertical and more out of the bar path. Your legs move gradually out of the way as the bar raises.
If you are using shoes with a heel it could be pitching you forward with the weight on the balls of your feet. You could try deadlifting barefoot or moving to flat soled shoes. If shoes aren’t an issue think about initiating with knees first then hips. Be sure you are pressing with the middle of your foot and not off your toes. Some “cues” for this are to think about initiating the movement pushing the barbell off the ground with your legs rather than pulling the barbell off the ground.
If it doesn’t feel right for most people the sumo deadlift will eliminate a lot of shin contact and you may be more suited for that style.
Even with good form there is some contact and people commonly wear long socks, pants, or other protection.