In general, you cannot translate that one word in English with one word in Latin in all scenarios. Subordination in many languages tend to be very idiomatic, and both English and Latin are no different in that regard. (Think about: why "that" at all?)
Depending on the introductory verb, you typically wouldn't translate it at all. Instead, you use would use the infinitive + accusative construction, what's called indirect statement.
There are many rules around this, dictating e.g. how the tenses work, so you can't do a simple one-to-one with English.
So, for your example sentence, "I'm glad that it's X", you would use placere in the impersonal, the subject goes into the dative, and then you'd have an indirect sentence, just like in this example from Cicero's letters:
Mihi placet, si tibi videtur, te ad eum scribere...
I'm glad (lit. "it is pleasing to me), if it seems good to you, that you write to him...
Other possibilities that eventually took over Latin in later eras are quod and quia or quoniam. The former seemed in earlier Latin to mean more "the fact that" or otherwise used when introducing a factual statement, while the latter originally meant "since" or "because", but by later Latin (and in earlier, uneducated speech), they could both be used to simply introduce an indirect statement. Wikipedia has a quick breakdown.
So using gaudeo (thanks to Cerberus for the suggestion), you could do gaudeo quod..., but in Classical Latin would mean something along the lines of "I rejoice in the fact that...", which is only a short hop away from "I'm glad that...".