I feel the steering wheel and the front wheels wiggle when I am at a speed 40-50 mph and I hit the brakes. The wiggling or shaking is worse when I’m at 70 mph and I hit the brakes. What’s wrong with it? How much would it cost to fix it?
The first suspect would be warped brake rotors and depending on the vehicle mileage, this could also be caused by worn/loose wheel bearings or a worn suspension component such as a tie rod, tie rod end, ball joint, etc.
No idea on the price as that can vary so much depending on the locale, shop rates, parts source, etc.
Assuming this is a front disc brake rotor problem (normal actually) you should figure on a new set of pads and new rotors.
Price it at 4 or 5 places (avoid the chains such as Midas) and that will give you a good average.
It sounds like it is time for new brake pads and rotors. This is basically maintenance. Most modern cars have efficient light weight rotors that are considered wear items like the brake pads. If you get them hot once they can do what you are feeling. Cheap ones tend to do it even more often. In the old days they were heavier and thicker so often they were machined and reused.
Try a few hard (not lock up) stops from 70 and see if it goes away, it always does for me.