Let’s take this from the top:

1. The United States Constitution is the supreme law of the United States of America per the United States Constitution.

2. The 14th Amendment is part of the United States Constitution including the “privileges or immunities” clause.

3. Privileges or immunities is constrained by judicial jurisprudence post-Slaughter House which basically interprets it to mean that the States will not infringe on the privileges or immunities (not to be mistaken for privileges and immunities) that US citizens have as US citizens. You might be thinking the first 8 amendments of the Bill of Rights, but the joke would be on you because those were incorporated via the Due process clause.

Unfortunately nobody really knows what the privileges or immunities clause was for except as a worse reading of privileges and immunities per Article IV because this is a Civil War amendment, like the 13th and 15th, and unlike the founding era amendments (excluding the 11th), the Civil War amendments were written at a time when lawmakers were all feeling a little bit French and a little bit Fancy with their legal prose because this was now a post-Napoleonic Code world. Clarity and brevity were not the high points of this era of lawmaking, but at least we had mostly started to standardize spelling and grammar.

What it boils down to is you have the right to travel, including take up residency and citizenship in another State. If there is anything else it protects, the Supreme Court does not appear to have found it yet.

So when you say:

> But the Supreme Court definitely has the power to check states when they curtail individual liberties.

You’re not actually contradicting me because as you quoted:

>> The Court does not exist to “check” the States except where the laws of States conflict with the laws of the United States, in which case either party may be “checked”.

This includes the US Constitution.

SCOTUS is governed chiefly by Article III, and here from Section 2:

> The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority;—to all Cases affecting Ambassadors, other public Ministers and Consuls;—to all Cases of admiralty and maritime Jurisdiction;—to Controversies to which the United States shall be a Party;—to Controversies between two or more States;—between a State and Citizens of another State; —between Citizens of different States, —between Citizens of the same State claiming Lands under Grants of different States, and between a State, or the Citizens thereof, and foreign States, Citizens or Subjects.

That’s their basic jurisdiction. The Congress has established inferior courts, so SCOTUS typically acts as a final court of appeal rather than taking original jurisdiction over cases, but there are cases where they can take original jurisdiction (arguably must, but well that varies by Court era).

In other words, they take cases in law and equity according to the laws of the United States. Not one word in there is about “checking” or “balancing” State power except pursuant to the laws of the United States. If a State infringes on someone’s rights, it is not under the jurisdiction of SCOTUS unless it is also a violation of United States law. That is an important and very easy to miss distinction, but not irrelevant. We have a lot of laws with a lot of “rights” that Congress has passed per Article I, the Bill of Rights and also the 14th Amendment, but the important thing is they either have to be in the Constitution or in a law which Congress has passed in order for it to be a controversy which the Supreme Court or the inferior Federal courts could take jurisdiction over. Without a case to substantiate it, SCOTUS does not have wholesale power to “check” the States because “checking” power is not what they do: they arbitrate disputes and pass judgements in accordance of laws.