Why is it normal for a solid to conduct heat even quicker from a heat source(e.g. flame) than a liquid or a gas?

a lot fewer molecules per unit volume (and thus distance) that need to gain energy.  Thus, the energy spreads over a larger volume per unit time in a gas relative to a liquid or solid.

Also, the direct contact interaction (which is conduction) can occur over much longer distances because of the low density of the gas.  That is, the distance to closest neighbor is much higher in a gas than in a liquid or solid, and conduction occurs when one molecule interacts (transfers energy) to another.  The reality however is that there is more convection than conduction when dealing with a gas when compared to liquid and solid, as a general thing.

actually, under the right conditions, liquid will transfer heat faster than a solid. Look up "heat pipe".

A heat pipe is a heat-transfer device that combines the principles of both thermal conductivity and phase transition to effectively transfer heat between two solid interfaces. At the hot interface of a heat pipe a liquid in contact with a thermally conductive solid surface turns into a vapor by absorbing heat from that surface. The vapor then travels along the heat pipe to the cold interface and condenses back into a liquid – releasing the latent heat.

How fast conduction occurs depends on the tightness of the particles in the material.

There is more space between molecules in a liquid than in a solid so conduction happens slower. Of course, for transport of thermal energy in liquids we have convection.

In gases, the vast emptiness between molecules makes conduction terribly slow but it's exactly why convection happens so rapidly.