No - there exist finitely generated torsion groups with unbounded exponent.
The Burnside problem asks if there exists a finitely-generated torsion group. Golod and Shafarevich proved that there does exist such a group, but the group they construct has elements of unbounded order. Novikov and Adian proved that there exists such a group of bounded order, while Ol'shanskii proved that there exists groups where every proper, non-trivial subgroup has order $p$ for some fixed prime $p>>1$ (such groups are called Tarski Monsters).
The links are all from wikipedia, as the original papers are all in Russian. Ol'shanskii wrote a book containing his proof, which you can try and read (although it is pretty close to unreadable...). It is called "Geometry of Defining Relations in Groups".
EDIT: I should say that we don't have to reach as far as infinite groups. For example, the group of symmetries of an icosahedron can be generated by and element $a$ of order two and an element $b$ of order three. However, $ab$ has order five. Of course, then one can take $a, b, ab$ as a generating set, or more generally the whole group as a generating set, to side-step this issue. One cannot do this with the infinite torsion groups I mentioned.