In sp² hybrid orbital, why does the p orbitals only have 3 sides instead of 4?
First, an $sp^2$ orbital does have two lobes. One is large, and one is tiny. Secondly, $sp^2$ means "a 'mixture' of one $s$ and two $p$ orbitals". Like the following:



So, you plug in three orbitals, and you get out three more orbitals. When drawn in place, you get this:



There are a total of six lobes here, but they are distributed amongst three $sp^2$ orbitals.

The number of lobes need not stay constant--it's the number of orbitals that does, since each orbital corresponds to two electrons, and there's no way for new electrons to be added/removed (yet). What's happening is that the wavefunctions(orbitals) of the electrons are "mixing up" to distribute the electrons in a way that stimulates bonding (one per orbital). The lobes have a significance, but it's much deeper.