Every few weeks someone asks how to buy the next house without first selling this one. There are three answers, and each has a price.
Bridge financing
You borrow against the equity in the current house to fund the deposit on the next. Fast and clean, and the most expensive of the three once you count the arrangement fee and the months of double interest.
Seller leaseback
You sell first, then rent your own house back from the buyer for thirty to sixty days. The cheapest option when it is available, because the buyer is usually happy to take the rent. It depends entirely on finding a buyer who is not themselves under time pressure.
Contingent offer
You make the purchase conditional on your sale completing. Costs nothing directly, and costs a great deal competitively: in a market with multiple offers, a contingent one is read as the weakest in the pile.
In a balanced market, take the contingency. In a tight one, it is usually cheaper to borrow than to lose the house.