Right now, a lot of Bitcoin is held at exchanges, which you can consider an IOU. However, it's still real Bitcoin reflected on-chain. As a user, you basically bought your Bitcoin from the supply of real Bitcoin the exchange holds in aggregated wallets. Should the exchange run short in supply, they buy real Bitcoin from other parties. It's all on-chain. Exchange wallets are well known and actively tracked.
Theoretically, an institution could let you buy Bitcoin that doesn't exist or which they don't own. This strategy would fall apart in mere weeks.
Say there's a new bull run and people are mass buying Bitcoin, for about 50 billion USD per week. You, the exchange, don't actually have this Bitcoin, yet in your books you pretend that you do. The customer is none the wiser. Your revenue is 50 billion USD.
Bitcoin doubles in USD value and customers are now mass selling and/or withdrawing their Bitcoin (which you made up). So now you have to either payout 100 billion USD or actively buy real BTC at its high price, which just doubled. Unless you have an additional 50 billion USD at hand, your exchange collapses and goes bankrupt.
Can an exchange sell slightly more BTC than they own? Technically, yes. Can they do many multiples of actual supply? No.
What you describe is a currency exchange risk, which would be the same of a bank holding a position in (for example) Japanese Yen but not covering it and holding their assets in USD instead.
The fraction reserve banking is a bit different issue, it's mostly about working in a single type of currency but backing short-term liabilities (e.g. account balances) with long-term assets (e.g. mortgage loans). So a customer A deposits 1 BTC and on-chain it gets transfered to the institution. Then when a customer B gets a loan of 1 BTC there's still just 1 on-chain BTC but 2 BTC-denominated IOUs, so the effective supply has doubled. Obviously, both of them can't withdraw the same coin at once, but the institution is not bankrupt as it has enough assets to cover all their debts, it may be temporarily insolvent though if a bank run happens.
The key difference from ordinary banking, of course, is that noone really takes loans denominated in BTC, so right now there's not much that an institution could have on the asset side other than "real" BTC. However, in a hypothetical world where BTC becomes the "mass market money" in some countries, then there would be BTC-denominated loans which would (among many other interesting effects) drive the pressure towards an equivalent of fractional reserve banking.
A clarification: IOUs are not money, bank deposits are. The money supply is the money issued by the central bank (aka monetary base) plus bank deposits.
Theoretically, an institution could let you buy Bitcoin that doesn't exist or which they don't own. This strategy would fall apart in mere weeks.
Say there's a new bull run and people are mass buying Bitcoin, for about 50 billion USD per week. You, the exchange, don't actually have this Bitcoin, yet in your books you pretend that you do. The customer is none the wiser. Your revenue is 50 billion USD.
Bitcoin doubles in USD value and customers are now mass selling and/or withdrawing their Bitcoin (which you made up). So now you have to either payout 100 billion USD or actively buy real BTC at its high price, which just doubled. Unless you have an additional 50 billion USD at hand, your exchange collapses and goes bankrupt.
Can an exchange sell slightly more BTC than they own? Technically, yes. Can they do many multiples of actual supply? No.