When doing dumbbell bench press, you feel that your triceps is the weakest link among all muscles involved. So at first glance, attempting to strengthen your triceps with isolation workouts in order to get it up to par with the other muscles seems like the appropriate thing to do.
It most probably is not. Judging from the numbers you gave in your other posts, I think it's safe to categorize you an 'Advanced Beginner'*. As such, dedicating an entire workout to triceps exercises is counterproductive. You're just not there yet, your body (the triceps specifically) can't handle the volume: you're overreaching. So when you do the dumbbell bench press on Tuesday, your arms are probably still struggling to recover from the unusually high load of saturday.
Consider this: When you're doing dumbbell bench press, the triceps are the muscles that fail first. This means that from all muscles involved in the movement, the triceps received the highest training stimulus. So, instead of doing a dedicated arm day, just go for another chest session on Saturday. Maybe the triceps will eventually grow proportionally stronger until another muscle will be the weakest link, maybe it will always be the tricep that is lagging behind. Whatever it will be, it shouldn't matter to you until you have several years of structured training under your belt. If anything, it would be more sensible to do an isolation exercise for the pecs and front delts on the same day in order to have them receive a training stimulus equal in magnitude.
TL;DR: Yes, it is normal. In compound movements, there will always be a weakest muscle. Instead of isolating that weakest link, just continue doing the compound movement.
*: 'Beginner' in terms of how your workout should be structured according to Mark Rippetoe's work