It's very hilly around here and I regularly get up to 45mph, with mountain rides topping 50.
My bikes are steel and feel perfectly solid at high speeds. I adjust air pressure and inspect tires before every ride, and give my brakes a good stop test before getting up any speed. It's a habit that comes from many years of motorcycling and bicycling. Your tires and brakes are the most important parts of the bike when it comes to staying upright.
Az