I would guess this is some standard fact related to the zerofree region. But cannot find it in the textbooks I read.

3$\begingroup$ Upper bound: Summation by parts. Lower bound: use the 341 inequality. $\endgroup$ – Fan Zheng Dec 3 '15 at 18:50
For starters, see Montgomery & Vaughan's "Multiplicative Number Theory", Theorem 11.3 and 11.4. For example, in the zero free region of 11.3, $$ \frac{1}{L(s,\chi)} \ll \log(q(t+4)). $$