Is cycling in the Netherlands without a helmet safer than cycling with a helmet in the USA?
There's a simple answer to the question you didn't actually ask: in the Netherlands cyclists who wear helmets are much more likely to be hospitalised than those who don't

Although the Netherlands is probably the safest country in the world for cycling, helmet wearing among Dutch cyclists is rare. It has been estimated that only about 0.5 percent of cyclists in the Netherlands are helmeted.
However, according to Dutch Government data (Rijkswaterstaat, 2008), 13.3 percent of cyclists admitted to hospital were wearing helmets when they were injured. Why does wearing a helmet appear to increase the risk of being injured so substantially?
The answer is probably related to another statistic. Of the injured cyclists wearing helmets, 50 percent were riding mountain bikes and 46 percent were riding racing bikes (Rijkswaterstaat, 2008). In other words, most helmeted cyclists in the Netherlands are engaged in a competitive activity, with very few making utility trips on the traditional style of Dutch bicycle.

I can't find a useful answer because of that. I couldn't find statistics that answer &quot;of the less than 1% of utility cyclists who wear helmets, how many were killed or injured&quot;?
Of course, having so many people riding bikes on the roads also means that of the people killed on the roads, a high proportion are cyclists (about one third). The Netherlands is also the safest place in Europe to use the roads (45 deaths/million pop, cf 145 in Greece which is the worst in Europe and 147 in the USA).
What that means is that in The Netherlands about 185 cyclists die every year compared to 700-odd in the USA (~4x as many, thanks whatsisname). There are more than twenty times as many people in the USA... but more trips by bicycle in the Netherlands. You're still safer there than any inhabited part of the USA. You're safer on a bike in The Netherlands than in a car in the USA, even just looking at immediate crash danger rather than life expectancy (the lifetime risk of death remains 100%, obviously).
There are some amusing results of that:

After a recent decline in the murder rate, you are now fractionally more likely to die while biking than to be murdered in the Netherlands!

In Amsterdam, you're still more likely to be murdered, though.

You're also more likely to die by murder in the U.S. as a whole than by biking in the Netherlands.

You are also more likely to drown here than either die biking or be murdered, especially if you are a child.


My suggestion is to look at it the other way: if you-the-parents insist that your daughter wear a helmet while cycling, will she keep cycling? Will she be unhappy that her mother would prefer her be socially ostracised than even try to fit into the new culture? Will she just observe that her parents are failing to assimilate and reject them, rather than everyone she knows?
Also try flipping it - how would your wife deal with someone who moved to rural USA and didn't want their daughter to learn to drive? Or insisted that she only travel in the Smart car, since that's the safest small car and it's what her parents had back home in the Netherlands?