In a medium pot set over high heat, bring 3 cups water to a rolling boil. Remove the pot from the heat, submerge the tea bags in the water, and steep for 5 minutes. Remove the tea bags and pour the tea into a pitcher along with 2 more cups of water. Cover and place in the refrigerator to chill.
Add the strawberry pieces to the same pot and mash with a potato masher (or fork). Add the remaining 1 ½ cups water to the pot, stir to combine, and bring to a boil, stirring occasionally. Reduce the heat to a simmer and cook for 15 minutes, stirring occasionally.
Set a fine mesh strainer over a large bowl and strain the strawberry juice into the bowl, pressing on the strawberry pulp to ensure that all of the juice has been extracted. Discard the pulp. Return the strawberry juice to the pot along with the sugar. Stir until the sugar is dissolved and bring to a boil again. Reduce the heat to a simmer and cook for 5 minutes, stirring occasionally and skimming off any foam that may appear.
Remove the strawberry syrup from the heat, allow it to cool for a few minutes, and then pour it into the tea. Cover and chill. Just before serving, taste the tea. If weaker tea is desired, stir in additional water, ½ cup at a time, tasting in between additions. Pour over ice and serve immediately.
Be sure that the pot you use is not too small, as the strawberry syrup will foam and rise up the sides of the pot while cooking. I use a 3-quart pot and it works perfectly.
If you prefer, you can make a pitcher of tea and a batch of strawberry syrup and keep them separate, allowing people to add their desired amount of syrup to their glass of tea just before drinking it.