```python
from typing import List

def steer_forklift(aisles: int, bays: int, moves: List[str]) -> List[int]:
    if aisles <= 0 or bays <= 0:
        raise ValueError("Aisle and bay counts must be positive integers.")
    
    for move in moves:
        if move not in ["north", "south", "east", "west"]:
            raise ValueError("Invalid move word.")
    
    for move in moves:
        if move == "north":
            bays += 1
        elif move == "south":
            bays -= 1
        elif move == "east":
            aisles += 1
        elif move == "west":
            aisles -= 1
    
    if aisles < 0 or bays < 0:
        raise ValueError("The forklift would carry it off the floor.")
    
    return [aisles, bays]
```