You can attempt to put in a buy offer contingent on selling your house. We did that with the house we're in now. Closed in one day on 2 houses. Morning, did the sale closing while my friends finished loading the moving truck. Went home with pizza for everyone, closed the truck and drove to the new house while the lawyer did the buy closing.
Once listed, our little 60's ranch starter house sold in 3 weeks.
I have had many friends take on a bridge loan even with offers on their houses. They all thought they knew more about the value of their houses than the real estate agents telling them that's all they're going to get. All were stuck with 2 mortgages for between 3 and 5 years. All of them took a huge bath when they finally did sell.