Similar situation and similar age here. It has gotten better with time. We remain factual and calm, and never dismiss their experience of pain. Hiding the pain of a procedure from a child makes it worse, IMO, and decreases trust, so we work with dealing with the pain if and when it occurs (ie: wriggling toes, counting aloud to twenty, roaring like a lion when it gets bad and, of course, distraction).
We always explain that it needs to be done and DC can make as much noise as they want/cry when the blood is being drawn/cannula is being inserted, but they do have to hold still. The more they wriggle, the more it hurts and the longer it takes. No kicking or hurting others, either. Definitely not allowed.
We have reached the point where we no longer have to hold DC down. We get through it easier every time now, which is great. There is hope!
Afterwards, DC gets rewarded with something small and inexpensive (eg: toy, lolly). They get to pick it beforehand if they think of it, or decide what they want during the procedure. If it is a particularly long and/or traumatising procedure, we go somewhere special after or do something special when they feel up to it (cinema, indoor play center, a McDonald's meal, visit to a special museum etc).
It is not easy, OP, but it does get easier once you find things that work for you. And don't discount that it is hard for you as a parent. Seeing your child experience pain and holding them day is awful. Reward yourself too after these sessions. Chocolate and a nice tipple at night ☺
Best of luck. Keep us updated to see how you go.