Are Turing machines more powerful than pushdown automata?
You can't go to the bottom of the stack without "forgetting" the rest of the stack. With a Turing machine you can easily go back and forth on the tape.

This simple task cannot be done with a pushdown transducer (roughly the same thing as the pushdown automata, but that can write things at each step), but can easily be done with a tape:


  Read a string $w$ and then write the string twice: $ww$


If you don't want to hear about transducers, the similar problem is for you: this language is easily recognized by a Turing machine, but not with a pushdown automaton:

$$L = \{ ww \mid w \in Σ^* \}$$

but I think with a transducer you get a grasp of the difference.