Why are gills typically external and lungs typically internal?

not positive but i am pretty sure that the gills only bring water in, a fish still has lungs and they need to be protected

Physical properties of water present four major gas exchange problems:
(1) higher viscosity of water ----> friction
(2) higher density ----> difficult to accelerate
(3) lower diffusivity of oxygen in water compared to air
(4) low solubility of oxygen in water
Gills are well adapted for gas exchange in a water environment in such a way that they work on principle of convective flow of water. The easiest way to achieve convective flow is to have inlet on one side and outlet on the other side i.e. one-way flow of fluid with blood flowing in counter-current direction to achieve best possible convection.
This system allows that blood leaving the gill have a PO2  greater than the PO2 of the water leaving the gill. PO2 (capital "P" with "O2" in index) means partial pressure of oxygen.

Atmosphere rich of oxygen have relatively high PO2. Getting arterial PO2 closer to outside air PO2 increase efficiency. To accomplish that, lungs have to be some kind of pressure regulator, or in other words to be "bag-like", with controllable pressure inside the bag.
The closed internal pool (or "bag" if you like) system of the mammalian lung provides for efficient gas diffusion exchange with an arterial PO2 that is close to, but does not exceed alveolar PO2.
On the other hand, birds lungs are internal organ because they are developed from lungs of land animals, but they apply complicated mechanism to achieve unidirectional flow (unlike bidirectional flow in mammalians). This is an adaptation to environment with less oxygen like atmosphere at high altitudes.