We know that the Bitcoin currency is volatile, but does the transaction fees correlate with it's volatility? Is it right to assert that if there is high level of Bitcoin demand, then the currency goes up, and then if the demand comes from a great amount of person, then the network size increases and so the transaction fees could be balanced by a greater availability of blockchain space, or am I going wrong ?
No, the more people use Bitcoin, the more the transaction fees will go up. This is caused by the fact that the blocks are generated with a 10 minutes average and they can only hold up to 1mb of data. The more people use Bitcoin, the more there's a demand to have their transaction in that small 1mb block. This is why there was a fork for Bitcoin Cash which advocates bigger blocks (Currently 8mb, but they will scale with the demand to keep fees low).
The logic behind the transaction fee in the bitcoin network is that the miners take in consideration first the txs that have higher fees to be added to the blocks, those txs are taken from the mempool, the place where they should wait until miners decide to add them to a block to be validated.
For example: If I perform a transaction that has a fee of 300 satoshis it will be first added in the mempool before it goes to a block; and let say, that some minutes later you perform a txs that has a fee of 30000 satoshis it will added to mempool too, but the miners will consider first your tx because of its higher fee.
Taking that in consideration, when the bitcoin price get up, usually there are a lot of people performing txs and since the size of the blocks is only 1MB, where only can be attached about 2500 txs, much of those people prefer to pay higher fees so their txs could be added to the next block. That cause the mempool get full of transactions waiting to be added to a block.
According that explanation we can say the transaction fees is not directly correlated with the volatility of bitcoin, it is more correlated with increase or decrease of quantity of transactions waiting in the mempool and the wish of people to get txs done asap. If there are many txs in the mempool people will be more tempted to pay more to get them done.