Learn Turkish: Bu mevsimde ormanda yaklaşık yirmi otuz mantar buldum. - I found approximately twenty or thirty mushrooms in the forest this season.

In Turkish, placing two sequential numbers next to each other creates an approximation meaning 'twenty or thirty' or 'twenty to thirty'. Adding a word like veya (or) is unnecessary for this kind of quick estimate.

Is it redundant to use both yaklaşık and yirmi otuz together?

It might seem a bit repetitive, but it is very common and natural in spoken Turkish. Yaklaşık just adds extra emphasis that this is a rough estimate. You could drop it and just say yirmi otuz mantar, and the core meaning would stay the same.

Why is it just mantar and not the plural mantarlar if they found twenty or thirty of them?

This is a fundamental rule in Turkish: nouns that directly follow a number, including numerical estimates like this one, must always remain in their singular form.

Why are there two words with the locative case right next to each other (mevsimde ormanda)?

They are simply providing the time and place of the action. Mevsimde tells us when ('in this season') and ormanda tells us where ('in the forest'). Turkish typically places time and place information near the start of the sentence, so stacking them like this is perfectly normal.