been told that if you travel more than 4 times in the same day the
oyster card automatically transforms your single charges into a daily
travelcard.

Regardless of how many times you travel, the Oyster card will work
out the cheapest combination of tickets that cover all the journeys
you have made and will cap your spend at that rate.

However, I have done so a couple of times and found out it
didn't happen and I had spent far more that buying a travelcard
without Oyster.

You should ring the Oyster card helpline on 0845 330 9876 and
either go through your usage with them or ask them to send you a
statement and check it yourself.

Is there a difference between using the tube or the bus in this
automatic system thing?

On the bus you only need to 'touch in' but on the Tube you must
'touch out' also. You will be overcharged if you only touch your
card at one end of the Tube journey.