Can this organization exploit the network by mining the majority of blocks to invalidate some transactions or in some other way?
"As pointed out before, giving rewards for creating blocks creates an arms race in computing power. Transaction fees only magnify the effect. This is very healthy for the security of our system — anyone can add blocks, so anyone can do it according to their own agenda, but to have any kind of significant effect on the whole thing they would have to invest a tremendous amount of resources.
Remember, the only way to determine which version of the transaction history is the commonly accepted one to look for the longest one. In theory, someone possessing 51% of the computing power of the network is able to completely control the system, as they can just start their own branch of the blockchain and create new blocks faster than anyone else. The resulting branch will become the longest and therefore accepted one, invalidating any blocks mined on other branches. Performing the 51% attack is however uneconomical even for actors the size of a government or a multinational corporation — it would simply cost too much for the trouble."
To give you some numbers, an Antminer S9 costs $3k and provides 14 TH/s. Current Bitcoin network hashrate is around 7,700,000 TH/s. Assuming a 51% attack is not underway already, an attacker needs to have at least the same hashrate as the rest of the network. This would cost over 1,5 billion dollars in non-reusable single purpose hardware alone. Add to this the ginormous power costs and additional costs of housing, maintenance, cabling etc. and you start to get the idea.
A sizeable entity who doesn't have complete majority can still make life harder for users by censoring specific transactions (simply deliberately choosing not to include them in their blocks), but this is more akin to a DDoS attack than a security breach.