So I understand the concept behind PoW (that if someone wants to overturn the current chain, they'd have to do all that work again).
But why is it even necessary since transactions are confirmed only if they contain valid signature?
Say there is Bob and Alice and miner X and all the other miners. Bob signs a transaction showing that he sent Alice $10, and the miners seeing the signature is indeed valid and there is enough balance, deduct it from Bob and add it to Alice. Now say miner x, wants to make a fake transaction to empty Bob's wallet for himself, sure, he could make a fake transaction with a fake signature, but in that case, all the other miners reject it, and he himself can't add balance to his own wallet from the empty air, because again, all the other miners reject it (another wallet with the actual balance has to sign the transaction). Now I can see how that would make rewarding difficult, but again, you can't just reward yourself without all the other miners also agreeing (could be that the reward goes to the first miner to add it to chain and again in case of conflicting chains, miners switch to the one with the most other miners in, the one that the majority agree on, and if they are both equal, this goes on until the majority switches to either chain).
And again, if someone just went and made up their own chain, all the other miners reject it, since they are only one miner, and the majority don't agree with that chain.
Now how do you track the majority? The same way you connect to other peers in the network, you connect to different peers, and always update this info (you connect to many peers, so that if one peer lies to you, most of the others tell the truth, you keep a list of all the other miners with you and the chain they all agree with).
You might then wonder "why can't one node create many clones and make you think they are many"? Sure, if they can overcome the sheer number of other miners that is and keep up (without getting blacklisted by other peers too, keep in mind that even without PoW there is still a cost). It'd be like acquiring all the ips on internet as an example.