Is it possible for a medium sized character to wield a large sized lance while mounted?
Can you please link FAQ.
No. A (sized for you) lance is normally a two-handed weapon. If you can wield it, you can wield it one-handed while mounted because of the weapon's property.
An oversized lance is unwieldable without some special ability. If you can't wield it, you can't take advantage of the weapon's property.