I think a bank run is caused when people believe their bank might be insolvent. A threat by masses of people to pull their money out of their banks simultaneously might cause a bank run. But that’s the power of the idea. The public has lost control of its own tax dollars. The only means it has to combat wall street is to threaten the government with an even worse scenario.
BTW, I would only support a bailout if the tax payers will make money out of the deal. In that case, it would be a win-win for everyone. Otherwise, I would oppose the bailout.