How were bastard children treated in the middle ages?

Based on my understanding, it somewhat depends on when in the middle ages you are discussing. In my period (the early middle ages) the ideas of primogeniture and of legitimate sons is not quite as cemented as it will be later on. For instance Charlemagne's first born son Pepin "the hunchback" was passed over despite being a "legitimate" son. That does not exactly answer your question but it provides some context. In the early middle ages ideas of legitimacy were not as secure. In terms of direct bastards, Arnulf of Carinthia is thought to be a bastard. His father Carloman may have married Arnulf's legitimate mother after he was born in order to legitimize the birth. Another example would be the issues faced by Charles the Fat before his downfall in the late 9th century. I'm at work so I don't have full access to everything but most info on that can be found in Simon MacLean's book Kingship and Politics in the Late Ninth Century Hopefully this is helpful!