“Think nothing of it” is a rare and very stuffy sounding expression in (at least my California dialect of) English. “It’s nothing” is better but still not too common.
As cced already said, common ways of expressing this in American English include “don’t worry about it”, “no worries”, “no big deal”, “no problem”, “you’re welcome”, “my pleasure”, etc.
Informal vs. formal and the gradient between them is always going to be an issue in any language, especially when comparing languages.
French also has more built-in structures for formal address/politeness, though many of them are being left by the wayside just as we left feudal lordships with a peasant class.
For what it's worth, I'd say that the more "stuffy" version of "de rien" in French would be "il n'y a pas de quoi".
As cced already said, common ways of expressing this in American English include “don’t worry about it”, “no worries”, “no big deal”, “no problem”, “you’re welcome”, “my pleasure”, etc.