You are correct that tRNA (and rRNA) is also synthesized by transcription. Once the RNA has left the nucleus, it folds up into it's appropriate shape. There are specific proteins (enzymes) that interact with certain tRNAs in the cytoplasm to add the correct amino acid.
Where do amino acids get attached to tRNA and where is it synthesized?
2 Answers By Expert Tutors

Jesse E. answered 06/05/19
Experienced Biochemistry Tutor
tRNA is synthesized by RNA polymerase III in the nucleus. Before it leaves, it goes through modifications.
One end of the tRNA is tasked with reading the mRNA. As the mRNA proceeds through this end, the opposite end of the tRNA is attaching amino acids. Once the mRNA is read, a corresponding amino acid is added to a growing poly-peptide chain. The growing poly peptide chain is located on the opposite end of the tRNA. I will attach a link in the comments that has pictures and further explanations of this process.

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Jesse E.
https://www.ncbi.nlm.nih.gov/books/NBK26829/06/05/19