"Why do we swear on a Bible in a court of law? You know, "I swear to tell the truth so help me God". Is a court of law no longer a government institution?"
We don't. In no court in the USA are you ever required to use a bible or say "so help me God," in any oath. You're completely mistaken. You may always simply affirm your intention to tell the truth.
Too many Perry Mason shows, I'm guessing.
When in the course of this nation did that change? Because that used to be the oath taken and a real Bible of OT and NT was placed under the left hand while raising the right hand ... I did this ritual many years ago in a court just as I have described it. What influenced the change to no longer use the Bible or require the fealty? And without the fealty then you ahve a self-fulfilling loop where only threat of being discovered in the here and now prevents lying to achieve whatever goal one has. Do you see the difference in what is at work in the human value system when that fealty to your Cretaor is removed for the purpose of 'separation of church and state'?