No need to taper off of Lexapro first?

I’ve been on Lexapro for about 6 months: 5 months at 10mg and then the last 3 weeks at 15mg. Unfortunately the side effects have not gotten better for me so I’m switching to Prozac (20mg). My doctor said I don’t need to taper off Lexapro first since they are in the same class of drug— I can just stop taking the Lexapro on the day I start taking the Prozac.

I’m inclined to just trust the direction of my doctor, but I’ve read a lot on here about people tapering too quickly or not at all and having some adverse reactions. Any thoughts?