A brake shudder that is barely noticeable around town but shakes the car when you brake from 60 mph is one of the most common brake complaints. The speed connection is a clue to what is happening. Here is the cause and the fix.
Why it is worse at high speed
Brake shudder is a cyclic vibration tied to the rotor - as the rotor spins, an inconsistency makes the pads grab harder and softer many times per revolution. At higher speed the rotor spins faster, so those pulses come faster and with more energy, turning a mild low-speed wobble into a pronounced highway shudder felt in the wheel, pedal, or seat; see what is brake judder.
The usual cause: thickness variation
Most highway braking shudder is disc thickness variation (DTV) - the rotor is very slightly thicker in some spots than others, often from uneven pad deposits or wear rather than true heat-warping; see warped rotor causes and pedal pulsation causes. Runout from a dirty hub or uneven lug torque contributes too - see new rotors vibration.
Where you feel it
A shudder felt mainly in the steering wheel usually points to the front rotors; a shudder felt in the brake pedal or seat often points to the rear. This helps narrow down which rotors are the problem before you start.
The fix
Mild deposit-based shudder can sometimes be cleaned up with a hard re-bedding, but established thickness variation means resurfacing or replacing the rotors - see rotor resurfacing cost. Fit rotors as a pair with quality pads and fresh hardware, clean the hub, torque the lugs properly, and bed them in.