No. Taekwondo is a different martial art to Karate.
Taekwondo and Karate are related. Many of the early Taekwondo practitioners had trained in Karate under notable Karate practitioners, and they incorporated some of the principles of Karate into Taekwondo, including the idea of forms, the training clothes, and the belt system.
However, Taekwondo and Karate are sufficiently different to be widely considered separate martial arts. Taekwondo emphasises kicking techniques more. The rules of sparring are different; Taekwondo uses a completely different set of forms; and Taekwondo follows Korean cultural traditions rather than Japanese.
There are multiple distinct styles of Taekwondo – just as there are multiple styles of Karate. The two most popular styles of Taekwondo are: Changheon-yu Taekwondo (often called ‘ITF Taekwondo’) and Kukki-won Taekwondo (often called ‘WTF Taekwondo’ or ‘WT Taekwondo’).