Nope. They drove them back during the battle for the city and then used a captured bat to drop some of their explosives on the forsaken forces and tanks in order to buy the Gilneans enough time to evacuate onto the Night Elf ships. Then in Silverpine Forest the Gilnean Liberation Front fought them all the way back to the wall but ultimately surrendered when Sylvanas threatened to turn Lorna into a Forsaken if Crowley didn't cooperate. Then a black dragon came and hung out for awhile and now it's abandoned.
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Arthas killed Ner'zhul inside the helm and was in complete control for awhile. The sword and helm corrupted him but he was by no means under their control. He even cut out his heart to rid him of the last of his humanity.
TBC was just poorly written. It was supposed to be an expansion about the Burning Legion but turned into an assault on an anti-hero creating an army to battle the Burning Legion and multiple assaults on the same Blood Elf prince. I guess Kael'thas did eventually align with the Burning Legion, but overall that expansion was a mess and Blizzard has said they regret the decisions made then.