I reached the point where I really needed to research 'me alegro de que' and 'me alegra que'. (I realise they are both correct, sorry I didn't make that clear.)
By and by I found this, which I hope might be timely for some other learners.
I've added Bill's suggestion, here's the link RAE Diccionario panhispánico de dudas
That's a good website, although I also suggest that you check online the 'Diccionario panhispánico de dudas', published by the RAE, to solve those doubts. There is a good explanation about 'dequeísmo' too, although it may be a little dense for some people Dequeísmo
By the way, I don't know if you did it on purpose, but you compared two correct expressions, because in the second you used the non-pronominal version of that verb, even though there is a pronoun.
'Me alegro de que..' is correct
'Me alegra que' is also correct
'Me alegro que' is incorrect, though.
'Me alegro de que pienses eso': the subject is 'yo', and the 'suplemento' or 'complemento de régimen' is 'de que pienses eso'. That preposition can't be removed, because this meaning of 'alegrar' is pronominal, and doesn't take a direct object, but a prepositional one, called 'complemento de régimen'. If you do remove it, then you incur in 'queísmo': 'Me alegro que..' (wrong).
'Me alegra que pienses eso' is correct too. The key here is that 'alegrar' is conjugated in the 3rd person, different than the pronoun (1st person), so it's not used as a pronominal verb, but as a regular transitive verb that takes a direct object, not a prepositional object. The subject is 'que pienses eso', and 'me' is the direct object. What would be wrong here would be saying 'Me alegra de que..' (dequeísmo).
Here's a wikilink with a list of similar verbs: