Bike wobbly at speed and falls into turns. Normal?

Damaged or overtightened steering stem bearings can cause all sorts of handling ills as well.

I copied this from a post from Jeff Saunders (owner of Z1 enterprises)

Motorcycle Wobble Causes

There's a LONG list of potential area that can cause this.

These are not in any particular order

Wheel bearings

Steering bearings - repack grease, check tightness

Swing arm bushings/sleeves/bearings - replace if worn, repack grease

Fork springs aging - check spring length

Unequal fork oil - if seals leak, then your damping can be ineffective

Rear shocks aging, damper leaks - if you have oil leaking from the seals, the damping is toast.

Worn Tires/Mismatched tires/tire pressures/wrong size tires

Wheel alignment - don't trust the chain adjusters - I visually align the wheels.

Bent rims/broken spokes - get the wheels off the ground and spin them - check for runout / damage

Front disc warped - check for runout

Steering & swing arm bearings are very common problems, but so are tire issues (pressure/mismatch) and wheel alignment.