Juzgar is obviously related to juez, the word for judge, so the first translation that comes to mind is "to judge." In certain contexts, that is the right solution. Take this verse from the Bible: No juzguéis, para que no seáis juzgados. This is translated as "Judge not, that ye be not judged" (Matt. 7:1). But when the word following juzgar is a "person" or a "case" the correct translation is "to try." Será juzgado por dos delitos means "He will be tried for two crimes" not "He will be judged for two crimes." Similarly, La causa será juzgada por lo penal means "The case will be tried in criminal court."
Another thing to note is that adjudicar never means "to adjudicate." Adjudicar un contrato al mejor postor means "to award the contract to the highest bidder." To say "adjudicate" in Spanish we use the verb resolver: La causa será resuelta mañana. The case will be adjudicated tomorrow.