Consider verifiers in the usual sense:

Many lecture notes on PCP say the following: Adaptive verifier with q queries is equivalent to non-adaptive with some q' queries where q' is a constant and both using same randomness. In other words I think this means

Non adaptive q' queries can simulate adaptive q queries. But I don't see why? How to show this?

Consider an adaptive algorithm making $$q$$ queries. Fixing the randomness, we can describe the positions queried by the algorithm using a function $$A$$ which gets as input the answers of the preceding queries, and outputs the next position to be queried. If the bits are read from an array $$x$$, then the adaptive algorithm works as follows:
1. Read $$b_1 = x_{A(\lambda)}$$.
2. Read $$b_2 = x_{A(b_1)}$$.
3. Read $$b_3 = x_{A(b_1b_2)}$$.
5. Read $$b_q = x_{A(b_1 b_2 \ldots b_{q-1})}$$.
To simulate this non-adaptively, we simply read $$x_{A(w)}$$ for all words $$w$$ of length at most $$q-1$$. Given these $$q' = 2^q-1$$ non-adaptive queries, we can simulate the adaptive algorithm.