That is completely inaccurate. Most of the Roman legionaries fought with swords, which required each legionary to have some 'personal space' in which to wield his weapon. This is in contrast to the traditional Greek phalanx formation, where the tactic of creating a wall of shields and spears gave very little space to each individual soldier.

In close combat, Roman legionaries would effectively fight one on one against an enemy soldier. There would basically be an enormous series of duels between individual soldiers[*] along the front line of the legion, with soldiers swapping in and out of combat as they became exhausted.

I have recently written a more detailed description of Roman battle lines here, if you are interested in more details and sources.

Edit: I forgot to comment on the testudo (turtle formation) bit. That formation was used not in pitched battles but in assaults on fortified places. The testudo provided protection to the assault troops on the approach to the fortification; it was not a formation for fighting.

---

[*] Source for that specific statement:

Gregory Daly, Cannae: The Experience of Battle in the Second Punic War (2002) p. 186

Source for the edit:

J. Thorne, Battle, Tactics, and the Limites in the West, in: P. Erdkamp (ed.), A Companion to the Roman Army, Oxford 2007, p. 218-266