You have two different problems.
First, the "mangled" output when using the 10X probes. As another answer has explained, the load of the 10X probe isn't enough to pull the output from the rectifier down to zero. A larger load across the bridge rectifier will take care of that. Even as little load as a couple of hundred kilo ohms would take care of this "problem."
Now to the more concerning case.
You have to realize that neutral and ground are connected together in your house, and that the ground clip of your oscilloscope is also connected to the ground of your house.
Your set up looks like this:
simulate this circuit – Schematic created using CircuitLab
Look over there to the left in the box labeled "Fuse box." The safety ground and the neutral are connected together.
Look over to the right at the box labeled "Scope." The safety ground of the scope is connected to the safety ground of the house. The scope safety ground is also connected to the scope probe ground.
Anything you connect to the probe ground clip is indirectly connected to the neutral of the house.
The junction of D3 and D4 is connected to neutral through your scope and the house safety ground.
That connection in effect causes a short to neutral on every other half cycle.
If you didn't have those resistors in there, you'd have a short circuit that would destroy the bridge and the scope and probably trip the circuit breaker - or maybe you'd be lucky and have a ground fault current interrupter (GFCI) in the house that will shut off the power to the outlet before anything gets damaged.
Another thing to keep in mind is that the ground clip of all channels of the scope are connected together. Clipping both ground clips to different points in your circuit that are not ground will cause a short circuit.
You have a similar problem to this question from last year.
This is the output of a full bridge rectifier where the scope and the source shared a ground:
The flat "bottoms" of your shorted bridge are slanted instead of horizontal, probably because of the 1M resistors playing with the capacitance of the circuit.
Be very careful when working with line voltage.
Things you don't know about (like the shared ground) can bite you really hard. Even things you do know but momentarily forgot can bite you. It only takes a moment's distraction, and you can end up connecting the scope ground to a hot wire - or worse, connect yourself to a hot wire.
If you must work on live line voltage, use an isolation transformer between the outlet and the device you are testing. It will prevent short circuits between the scope and the device, and between you and the outlet.