Is there an idiom that means to make big claims but have little evidence to support them?
But you can coin four-character words to express it more accurately. It looks like that you mean academic claims.
Let me have a try:
论訇证寡（The claim is thunderous(figurative), but the evidence is little.）
Note: Only in writing you can coin such words. Otherwise people won't understand you.
Not sure If 言过其实 can fit. it means a statement surpassing the fact.
or 夸大其词？ it means represent beyond the bounds of truth; make an overstatement;
This thread would never be complete without the "urban-dictionary" style (i.e, slang), modern example for such idiom, the good old
吹牛, or its R-rated version
It generally means to brag about something, which could be true but usually isn't.
In addition to some good answers: