"Do not worry too much."
Translation:너무 걱정하지 마십시오.
너무 is to a great extent, and fits the "too much" role of adverb. 많이 usually refers to something you can count, usually meaning many or "a lot of" in the role of adjective.
"너무 많이" in place of "너무" is now accepted. However, it is redundant so I would not say it's a good translation.
많이 is an adjective for" a lot/many" while 너우 is an adverb that means "to a great extent" fitting the role of "too much" in this context.
많이 means "a lot" but does not have the same meaning as 너무 which is "too much" or "excessive". So yes you could, if that is what you are saying. But in the case of ,"too much" 너무 is used