Yes, this is completely normal.
It simply means that there are more blocks being build ontop of the block your transaction was put in / originated from.
The more confirmations = less chance of your transaction being reverted in a 51% attack.
So my question is, how much exactly transactions I have to receive at the end to be sure it's confirmed? Because I sent my Bitcoin in the morning and now the night, but it still did not reach its destination and keep sending confirmations.
Usually 1 confirmation would be enough, ( to prevent double-spending, which to this day does still happen quite a bit.)
Theorethically 6 would be the most optimal. ( Unless there is a 51% attack.)
With six confirmations it is essentially mathematically impossible for an attacker with less than 51% of all mining capacity to get six blocks in a row. and still surpass the longest block chain. With 51% or a lot more than 51% the attacker can get six confirmations by creating a parallel blockchain in which only transactions approved by the attacker get included in blocks.