A Tribute Summon IS in fact a Normal Summon, and you can only Normal Summon Once per turn.
When Tributing for a Level 5 or higher monster Without a summoning condition, this is a Normal Summon.
You can Special Summon in a turn as many times as the opportunity arises.
In order to Special Summon you must first pay the cost or have the effect of another card active which summons the monster.
By pay the cost i mean, fulfill the summoning conditions in which the card needs to meet.
Exp:
Lets say you have an empty field, and your opponent controls a monster, you can Special Summon a "Cyber Dragon" from your hand.
Because the cards summoning conditions have been met.
If you have a Light & Dark monster in the graveyard, By banishing them, you will had paid the cost to Special Summon a "Chaos Sorcerer" from your hand within the same turn you Special Summoned "Cyber Dragon"
If you have the "The Tricky", you can pay its costs by discarding a card to the grave and special summon it from your hand, within the same turn you summoned both "Cyber Dragon" and "Chaos Sorcerer"
None of these care considered Normal Summon, so you still can Summon a monster from your hand. You can in fact use this time to Tribute summon as your normal summon for that turn, by tributing a special summoned monster.
Spell & Traps like Monster ReBorn and Call of the Haunted are also a special summon.
There are some cards that are Special Summon, Tribute Summon monsters, like Toon monsters.
Where you can keep tribute summoning in the same turn but these are still considered special summons.
By The Way, Any summon from you Extra Deck is Considered a Special Summon.
Hope This Helps.