[TheCoelacanth]

You don't need a permit to buy a handgun (i.e. the ones used for nearly all of the crimes) in the US to my knowledge.

[ThunderSizzle]

You don't need a permit to buy a gun, depending on the state, but you do need to pass a background check, show an ID, and if you wish to carry it outside your house, you probably need a permit, again depending on the state and whether you wish to conceal or open carry.

[TheCoelacanth]

37 out of 50 states do not require a permit for open carry.

30 out of 50 states allow purchasing a gun from a private seller without a background check.

There is simply no basis for the claim that US and Swiss gun laws are comparable.