Immigration can have an impact on wages, but It's a much more complex function than more immigrants equals more labor supply equals lower wages.
Immigrants also contribute to the economy. Depending on how much they contribute, immigration can raise or lower wages. The issue is how much do they contribute? Immigration doves argue they are a net gain, while immigration hawks argue they are a net detriment.
Historically, in the long run, immigration has always been a net gain.
I am not an economist but immigration rates and the relaxing of barriers of trade between nations would seem to be the most impactful policies on domestic wage growth. You can always argue that these policies have helped to lift the rest of the world out of poverty, but it doesn't belie the fact that increased immigration rates impact on wage growth for American workers. Maybe someone could shed more light on this issue, but I would suspect that both issues are inexorably tied.