It didn't start out that way, but it has become increasingly fashionable to abuse a certain clause, by ignoring the underlying logic.
Article VI says this about the "supreme law of the land":
This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
Treaties are listed in there. As being part of the supreme law of the land. Like the Constitution, itself. And the grammar could be read (by unscrupulous persons) as not requiring treaties to be made "pursuant to the Constutition". Think of it.
Some sort of Presidential memorandum of agreement with some official from another country could trump the Constitution!
Well, we would hope that no president would be that cavalier about his or her responsibilities. (Wouldn't we?)
The Constitution says elsewhere (Article II Section 2) that the president may make treaties, but that the treaties must be made with the advice and consent of the Senate, and must be approved by two thirds of the Senate present. A quorum, or more than half, must be present to do business. Surely that should be enough of a check on the power to make treaties?
Treaties can be long, complicated documents. Lots of funny stuff can be hiding in the fine print. I don't know about you, but when I hear people in the government trying to explain recent government operations of highly questionable Constitutionality (NSA? TPP? ACTA? ...), I hear funny things that sound like, "This is required by our agreements with other countries."
Would you argue that treaties should be able to overrule the Constitution?
Consider this: without the Constitution, there would be no legitimate government.
If a treaty could overrule the Constitution, a treaty, passed by one third of the Senate, plus one, could be used to dissolve the entire government and place the country under the authority of a foreign constitution.
If you think about it, the fact of the lesser requirement for passing a treaty should be understood to indicate that treaties were not intended to have greater precedence than even ordinary law, or even equal precedence.
Otherwise, a president seeking to force passage of an unpopular law could simply make an agreement with some foreign power and get a friendly Senator to push it through the Senate as a treaty when a lot of the Senate are absent.
The very suggestion that a treaty could overrule the Constitution should be considered treasonous words, in my opinion.
So, I would like to propose an amendment --
An amendment to limit the power of treaties.
No treaty or other agreement with foreign governments or powers shall be valid or of force except it be made consistent with the requirements of the Constitution and its Amendments.
No treaty shall be given greater weight or precedence, relative to the acts and operations of the United States of America, than the laws and Constitution thereof. No treaty may be used to void the Constitution of the United States of America or abridge the provisions thereof.
For any treaty to have weight or effect on the acts and operations or the United States or its citizens or other persons, regular laws must be duly passed implementing the provisions thereof. Without such implementing laws, no provision of any treaty shall be of force within the United States or any territories, possessions, or vessels thereof.
The judicial power of the courts of the United States is hereby formally recognized to extend also to the review of law and treaties for consistency with the law and the Constitution.
The courts shall not have power to revise any such law or treaty, only to nullify its effect and request the Congress to reconsider the laws and treaties so nullified.
The terms of this amendment shall not alter the effect of court findings under established proceedings of judicial review prior to ratification of this amendment.
[Updated in a separate post, to keep it readable.]