Not exactly. You say "I bet" when you "think something is or going to be true." You're not really sure about it, you just think that it is/will be the case. Example. I bet the students will be surprised with the lessons today. I bet he forgot to bring his wallet again!

They are different. I bet ... = It is probable that... I'm sure = I believe it is certain. You can say it to comfirm that you understand and agree. "It really hurt when I hit my knee in the football match." "I bet! It looked like it hurt!" Also, if you just say "I'll bet!", it means the exact opposite - it is sarcastic. "I'm really going to get down and study this time." "I'll bet! That is what you said last time and you got a D"