Say if I was creating a new bitcoin website and I want to transfer some bitcoins from one of my own private wallets to another would that make it so the transaction would confirm almost instantly because it is a closed loop?
Yes you're talking about an off-chain solution. Exchanges use this solution.
It would be prohibitively expensive and impractical to transfer ownership on-chain for small amounts and/or frequent changes, so you send your coins to a single wallet owned by the exchange.
The exchange has their own system to keep track of who owns what. This system is private and not transparent, but fast and potentially free. Then when you choose to send it from the exchange to your wallet, a network fee is incurred at that point.
The off-chain solution could also be a blockchain solution, though not distributed as no one else would have any interest in maintaining your businesses ledger.
e.g. with coinbase.com
The Deposit: Your wallet -> "your" coinbase wallet occurs on main net blockchain network fees incurred technically deposited into a single main net wallet wallet controlled by coinbase you trust them with your coins at this point Interwallet transfer on coinbase: "your" coinbase wallet -> another wallet on coinbase of "your"s occurs off the chain as such it's free as such is opaque could be implemented however coinbase sees fit: vb running in an excelsheet for all I know! Withdrawal: "your" coinbase wallet -> Your wallet occurs on main net blockchain coinbases single wallet is paying you network fees incurred grouped with other withdrawals for network fee efficiency coins in your full control again