I'm not aware of any recent studies, but the League of American Bicyclists (formerly League of American Wheelmen) has done some studies in the past that were reasonably convincing that helmets prevent serious head injuries and save lives overall.
But, interestingly, when they crunched the numbers they decided that the health effects of riding are positive (life-lengthening) even if you don't wear a helmet, so they have (at least in the past) not advocated for mandatory helmet laws. Better to not wear a helmet and cycle than to use the helmet as an excuse to not cycle.
But I haven't looked at any of this in 15-20 years -- since last I was involved Boy Scout cycling stuff.
[Incidentally, as to whether helmets save lives, when I was helping with Scouts there were two incidents where Scouts were thrown from their bikes and landed on their heads, in both cases with enough force to crack the helmet. There is no doubt that, without the helmets (that ARE mandatory for Scouting events) the boys would have been seriously injured rather than (as they were) just scratched up.]
[It should also be noted that the most injury accidents with bicycles are between a bike and some stationary object, or simply falls. Motor vehicle collisions are a relatively small fraction. Neither of the accidents I mentioned above involved a motor vehicle, and one was on a bike path. Helmets are needed even when not riding in traffic.]