I am trying to create a multiclassed druid/bard, but whenever I add levels of Wizard, regardless of how many, I can only see 0-1 level spells. 

To be clear: you have read and understood the rules about multiclassing: https://www.dndbeyond.com/compendium/rules/basic-rules/customization-options#Spellcasting

Can you provide a link to character sheet?

It seems to have just been a user interface bug. On the "Manage Levels" screen it would only display options to pick 0-1 level spells. But when I generated the character sheet it would let me add the appropriate spells when I went to "Manage Spells" under the Spells tab. After going back into the level editor it would allow me to see the appropriate spell levels again.

Usually I find once you change the class level in the builder (say changing Wizard from 1 to 5), moving to another section by clicking next or back, to either Ability Scores or Races locks the level change in so the spells section shows the correct capabilities.

I'm actually having the same issue.  I"m creating a Wizard NPC character and I can only see spells and filter them in the character builder up to Level 5.  After that I do not see spells at higher levels as options to pick or filter.

Quote from TheCarson >>

I'm actually having the same issue.  I"m creating a Wizard NPC character and I can only see spells and filter them in the character builder up to Level 5.  After that I do not see spells at higher levels as options to pick or filter.

 
Refresh page.

And ensure you're high enough level, for level 5 spells the character must be level 9 in a spellcasting class.