Would you agree that invaders and other unnecessary foreign labor is unnaturally depressing wages?
Lots of factors determine wages, including of course the number of people who are willing to perform a particular job, and their productivity and the price they are willing to do the work for.
People who are present in our nation illegally may be willing to work for lower wages than other workers, and there are some employers who will take advantage of that to reduce their costs. Needless to say they may also avoid other costs created by the government, since they are willing to violate the law.
I would agree that the failure of the government to enforce the immigration laws creates a number of adverse consequences, including the situation where businesses that comply with the law are at a competitive disadvantage in some markets. One example would be some types of construction work. That's one reason why the government should not simply decide not to enforce some laws, some of the time for their own political purposes.
An increased supply of equally qualified workers, willing to do a particular job, at a particular place, should decrease the wages necessary to hire the workers, if there are not other factors with a larger effect, and if the supply exceeds the local demand. That's a consequence of the effect of supply on pricing, and independent of where the workers happen to have come from.