Hi slow_rider
The problems you are encountering are common when driving displays.
As has been said by tggzzz in replay#4, 4MHz is not a slow signal, and especially when you take into account rise time, jitter, etc.
The other thing is that the TXB0108, and the resistor, resistor/BJT, resistor/MOSFET translators you see floating around on the net, are only good for slow signals, and even then they are best avoided as they do not meet the edge speed specification or logic level specification for MOS logic and fall apart hopelessly as the frequency increases due to parasitic capacitive loading.
You also need to decouple the chips with 100nF X7R ceramic capacitor across the supply pins.
It is not clear, but do you need bi-directional comms?