First off, it is not impossible. Bitcoin nodes are not validating thousands of transactions per second. They are actually processing less than ten per second, which is perfectly reasonable for even slower computers.
Second, not all Bitcoin nodes validate all transactions. What you are likely referring to are "full nodes", which are basically nodes that validate and relay transactions as they receive them. The network does not operate on a principal of how many nodes have validated a transaction. It operates on a principal of how much work nodes have done on top of a block containing a transaction. This work is not done by full nodes, however, but rather "mining nodes". These are the nodes that perform the proof-of-work algorithm on top of blocks, which are sets of transactions. If a mining node finds a nonce for a block that contains a transaction, that transaction is now fully confirmed. It doesn't matter how many other nodes have seen that transaction.
Of course, when the block is broadcast, other nodes will check the mining node's work to ensure that a block contains only valid transactions (transactions that are properly formed and don't double-spend or create new coins) and an acceptable nonce. This is a considerably less intense computational effort, and full nodes will perform this check prior to relaying a block. Since bad blocks aren't relayed, they won't propagate through the network, and miners don't waste their resources attempting to cheat the system.