Can I create a private bitcoin network? 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 -&gt; "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 -&gt; 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 -&gt; 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