Depends on your speed and the road surface. On rough roads, yes. On a smooth road at high speed, the higher pressure of a narrow tyre is beneficial, as is the reduced wind resistance.
It all pales into insignificance if the construction is poor, though.
I don't believe higher pressure (a narrow tyre should have higher pressure) increases punctures; normally it's a Good Thing as it reduces the risk of pinch punctures. Embedding of glass seems to be down to the hardness/stickiness of rubber used on the tread rather than anything else.
At equal pressure, should you for some reason want to do this, a narrower tyre has the same size contact patch, but it's a long thin one rather than a short wide one. The short wide one will hit more nasties.