  Given a graph $F$, we define $\operatorname{ex}(G_{n,p},F)$ to be the maximum
number of edges in an $F$-free subgraph of the random graph $G_{n,p}$. Very
little is known about $\operatorname{ex}(G_{n,p},F)$ when $F$ is bipartite,
with essentially tight bounds known only when $F$ is either $C_4, C_6, C_{10}$,
or $K_{s,t}$ with $t$ sufficiently large in terms of $s$, due to work of
F\"uredi and of Morris and Saxton. We extend this work by establishing
essentially tight bounds when $F$ is a theta graph with sufficiently many
paths. Our main innovation is in proving a balanced supersaturation result for
vertices, which differs from the standard approach of proving balanced
supersaturation for edges.
