The oversized tires could be placing a burden on worn front suspension parts, like worn ball joints or tie-rod ends. Also, if large tires are even the slightest bit unbalanced, this may cause the shimmy too. But, you said the shimmy happens when you brake. This may be indicitive of a warped rotor, which may not have been noticiable with the standard tires but now it may be projected ten-fold with larger tires. Bring the car to a shop that does both brakes and suspension repair and have it checked. A good tire shop will do both types of repair.
I have a 1999 Grand Cherokee. My son put oversize tires 275-45-R20 on it and now it has a violent shimmy when braking. It does not have it with the old stock size tires. Can it be the steering damper? Thanks for your help