Does my little cat have conjunctivitis or something worse?

My little cat has a swollen and watery eye. A veterinarian I know told me this is normal, since he's normally a barn cat and only spends time indoors. However, it's not getting any better, and he's leaking milky, yellow-green fluid from his eye. Not only that, but it's also very swollen.

Because he's so young, I was told he can't take antibiotics yet and that I should give him regular eye drops. However, I'm worried.