Synonyms for Mediate:

n.

cope, handle, shrug off, overcome, manage, carry off, surmount, work out, get through. bridge-building, buckle, concession, break up, call off. catalyze, inaugurate, start-off, make, invent, bring about, trigger, set off, create. arbitrate (noun)
propitiate.
mediate (noun)
in-between, intermediate, indirect, liaise, intercede, mediated, middle, arbitrate.

v.

compromise (verb)
compensate, appease, allow, adjust, concede, reconcile, bargain, settle, accommodate, compromise.
intervene (verb)
expedite, implement, aid.
judge (verb)
arbitrate, sentence, consider, referee, decide, rule, pronounce, find, assess, appraise, judge, adjudicate, determine, resolve.
mediate (verb)
intervene, assist, negotiate, intercede, facilitate, parley, conciliate.
order (verb)
group, devise, rationalize, unsnarl, plot, sift, fix, program, harmonize, maintain, normalize, establish, classify, array, schematize, class, subordinate, right, marshal, set, schedule, scheme, organize, plan, support, integrate, arrange, stratify, stabilize, rank, orchestrate, order, regulate, rate, separate, type, form, score, balance, shape, control, Methodize, chart, cast, unify, structure, place, screen, compose, collate, categorize, grade, prepare, frame, systematize, design, pigeonhole, sort.
try to bring to an agreement (verb)
intercede, propitiate, negotiate, resolve, intervene, conciliate, arbitrate, referee, intermediate, settle, reconcile.

Other synonyms:

pacify
propitiate.
Other relevant words:
cope, trigger, create, invent, buckle, manage, mediated, bridge-building, liaise, catalyze, inaugurate, intermediate, middle, make, handle, concession, overcome, indirect, in-between, propitiate, work out, surmount.