I had a similar problem during my South American vacation. I lost most of my infantry when I attacked the first northern city, I couldn't believe that English Armored Swordsman could be cut down by clubs and spears. Reinforcements were still many turns away but fortunately by this stage of the game my economy was such that I was able to hire an army of local mercenaries to supplement my meager cavalry & longbow men. These were inferior to the enemy's troops but they could be relied upon to pin them long enough for cavalry or infantry to sweep round and attack their rear. Please note this was with the game set at the default difficulty and I expect the harder settings would make the line route much quicker.