"Bet" is the usual word for "wager." If your dictionary doesn't show "bet," look up "wager."
"Bet" is a gambling term. Suppose you are about to watch a football game. "I just bet $50 that the Patriots would win the game" means that you made an agreement with someone just a few minutes ago. You said "If the Patriots win, you pay me $50. If the Patriots lose, I pay you $50." And the other person said "I'll take that bet. You're one!"
Phrases like "I bet X," "I'd bet X," "I'll bet X," "I just bet X" all mean the same thing. They don't usually refer to a real bet. They refer to an imaginary bet, a "hypothetical" bet. They mean "I am sure X will happen. I'm absolutely sure. I'm absolutely, positively sure. I'm so sure that I would be willing to bet money on it."
"I bet X" is normal conversational English, but, grammatically, it is not right. It should be "I would bet that X" or "I'd bet that X." Adding the "just" makes it even stronger.
"He promised he would wash the dishes. But, wait and see, I just bet he won't."
"Look at that man running to catch the bus! Do you think he'll make it?" "I just bet he won't. Look, the bus is already closing its doors."
"Luke and Leia are trapped in a giant trash compactor. How can they possibly escape?" "I don't know, but I just bet they will, because I know there are seven more movies in the series."