Learn Icelandic: Hefur þú séð nýja bæklinginn?

Sjá (to see) is a strong verb, meaning it doesn't form its supine by just adding a regular ending like -að or -t. While many strong verbs end in -ið in the supine (like farið from fara), strong verbs that end in a vowel in their infinitive, like sjá, often take the ending -ð. Notice that the root vowel also changes from á to é.

Why is the adjective nýja used here instead of the accusative form nýjan?

Because the noun it describes, bæklinginn (the brochure), is definite (it has the attached article -inn meaning "the"). When an adjective describes a definite noun, it must use the "weak" declension. For a masculine accusative noun, the weak ending is -a. If you were asking about "a new brochure" (indefinite), you would use the strong declension: séð nýjan bækling.

Why is the noun bæklinginn in the accusative case?

The verb sjá (to see) takes a direct object in the accusative case, because it is the thing being seen. The dictionary form of the word is bæklingur (a masculine noun). In the accusative case, the -ur ending drops, leaving bækling. Then, we add the masculine accusative definite article -inn to get bæklinginn (the brochure).