Shaking of the car under braking can be a sign of worn suspension bushings but is most likely a sign of warped front brake rotors. When you apply the brakes, brake pads come in contact with the brake rotor/disc in a very similar way as the brake blocks come in contact with the rim of a bicycle. If the rim is buckled you feel the brake lever on the handle bar pulse. Same thing happens when you apply the brakes in car, if the rotors are warped you will feel the brake pedal and steering shake and pulse on braking.
If the brake rotors are sufficiently thick they may be remachined and reused. If the brake rotors do not have sufficient material to machine away the rotors may need to be replaced.