Its the back of the bullet that is vaporizing from the start from the explosion and hot gases (since it has very low melting/vaporizing point), and then second, smaller spike is when hitting hard surfaces and melting instantly into 100s of pieces (if you watch any slow-mo impact of a lead bullet they mostly don't vaporize, just melt and fly parallel to surface being hit).

There is safe ammunition these days, but it costs more. Swiss make their military ammo lead-free (they have cca same ammo as US/NATO but bullets are most effective with different barrel twist compared to NATO ones, I think 1:7 vs 1:10), but both can fire each other's rounds safely.

Safe ammo is essentially just back of the bullet covering lead core (if present at all), but for some reason most manufacturers don't do it by default. Some stuff doesn't have lead core at all, but then desired weight needs to be achieved via other substances and is priced accordingly.