What evidence is there that quasars are small?
The fact that quasars are very luminous and change luminosity over short periods of time means that the must be small.
Quasars emit huge amounts of energy all across the spectrum. The luminosity of quasars varies over periods of days, weeks or months.
The change in luminosity means that the source of the energy must have a size which is smaller than the distance light travels in the change of luminosity period. They must be light days, weeks or months in size.
Quasars are caused my material falling into supermassive black holes at the centres of galaxies. These are millions to billions of times the mass of the Sun. Although massive, these supermassive black holes are typically smaller than our solar system.