Not only is it not normal, but its almost certainly a violation of the terms specified by the lender. The lender will state the seller can receive no benefit, and staying in the house three months is certainly a benefit. Look at the paperwork on the deal, and I think you'll find a statement to this effect. If not, ask for copies of any additional paperwork from the bank you'll be required to sign at closing and it will be in there somewhere. Better to find out now than to get to closing, be asked to sign this statement and to be unable to close.
My first advice would be to walk away if you can't have possession of the house on the day of closing.
If this is your dream house and you're determined to buy it and willing to put up with this BS, then the first step is to be absolutely certain the lender who's approving the short sale is fully aware the seller is staying. Add a statement to this effect into the contract, and push it back through the approval process. I believe this will actually kill your deal, but I suppose the bank could approve it.
If the bank does sign it, and you're willing to put up with this, make it a rental. Come up with a solid lease. Best to get one from someone in your area. You agent may be able to help. Set the rent and deposit high. The seller is wanting the benefit of not moving. You want to move in. Make them pay. At least make it somewhat higher than market rent. I'd be sure its well over enough to cover the payment on the new place as well as any expenses. I'd actually make it significantly higher than that. Require the deposit and first months rent (or, the full three months) at closing. Since I assume you're bringing some money to closing, it would reduce the amount you bring rather than giving you cash in hand.
Your lender is going to require that you qualify for both loans, if you're going to have to keep your existing place for a while. Is that a problem?
I'd plan to have to do an eviction. If they can't be out now, what's going to change in three months? Having a formalized lease gives you the basis for doing an eviction in three months when they refuse to leave. If you choose, you could let them go month-to-month and continue to collect rent, if they're willing to pay. My guess is that they won't pay and won't leave, and you have to do an eviction. That's a month long process that costs a few hundred in some places. If you've in a tenant friendly area, it can be many months and cost much more.
Seriously consider just blowing off this deal and finding another house.