What we did, and it seems to be the norm, is that a chain of houses is created. We put in the offer on our new house, subject to the successful sale of our current house. And the people purchasing our existing house were doing so subject to the successful sale of their existing house. All three property transfers happened on the same day.
There could be a number of properties in the chain, depending on if new or existing properties are being bought, and if someone renting is now buying a house.
Other alternative is to sell your house, rent short term (or move in with family), and buy for cash/mortgage. Sometimes the person selling the house will choose a lesser bid, if it is not part of a chain, as the closing can be cleaner, i.e. if one link of the chain breaks, for whatever reason, then all the sales/purchases in the chain fall down.