I remember reading about the wild magic zones witch I think in some places can be instilled to have random effects I can justify. I have a few ideas on how to implement them, just need to make or find a roll chart that I like lolI do not want to go into detail about whether or if you single out a wizard character, I believe you just ask for things that could be challenging for this character class and that is a fair question:
Counter spell, but do not overdo it.
Enemy spell casters can have shield also (think magic missile)
Also Dispel magic by enemy casters on enemy mobs which have been debuffed or restrained by the PC wizard.
More mundane (in a very fair way I think, I did it like that lately): Assume group of enemy mooks some melee some ranged and some commander. As soon as the PC wizard shows his danger e.g. casts a fireball decimating the enemy number, the commander reacts and orders his ranged soldiers to focus fire on the PC wizard. in that way he is allowed to shine but it has a price attached to it.
Some odd stuff: wild magic and magic dead zones (these were a nice gimmick coming up for FR in former editions, I do not know why this is not so popular anymore). Of course you need a justification for these to be there: For FR a former or damaged Mythal justifies the presence of these. For wild magic use any table also from former editions, or make up your own funny stuff like the caster farting butterflies, or a very cruel one (for some) changing gender.
Thanks for the great advice